Vibrational and optical properties of MoS2_2: from monolayer to bulk

Molybdenum disulfide, MoS2, has recently gained considerable attention as a layered material where neighboring layers are only weakly interacting and can easily slide against each other. Therefore, mechanical exfoliation allows the fabrication of single and multi-layers and opens the possibility to generate atomically thin crystals with outstanding properties. In contrast to graphene, it has an optical gap of 1.9 eV. This makes it a prominent candidate for transistor and opto-electronic applications. Single-layer MoS$_2$ exhibits remarkably different physical properties compared to bulk MoS$_2$ due to the absence of interlayer hybridization. For instance, while the band gap of bulk and multi-layer MoS$_2$ is indirect, it becomes direct with decreasing number of layers. In this review, we analyze from a theoretical point of view the electronic, optical, and vibrational properties of single-layer, few-layer and bulk MoS$_2$. In particular, we focus on the effects of spin-orbit interaction, number of layers, and applied tensile strain on the vibrational and optical properties. We examine the results obtained by different methodologies, mainly ab initio approaches. We also discuss which approximations are suitable for MoS$_2$ and layered materials. The effect of external strain on the band gap of single-layer MoS$_2$ and the crossover from indirect to direct band gap is investigated. We analyze the excitonic effects on the absorption spectra. The main features, such as the double peak at the absorption threshold and the high-energy exciton are presented. Furthermore, we report on the phonon dispersion relations of single-layer, few-layer and bulk MoS$_2$. Based on the latter, we explain the behavior of the Raman-active $A_{1g}$ and $E^1_{2g}$ modes as a function of the number of layers

Stacking effects on the electronic and optical properties of bilayer transition metal dichalcogenides MoS2, MoSe2, WS2, and WSe2

none3siEmploying the random phase approximation we investigate the binding energy and Van der Waals (vdW) interlayer spacing between the two layers of bilayer transition metal dichalcogenides MoS2, MoSe2, WS2, and WSe2 for five different stacking patterns, and examine the stacking-induced modifications on the electronic and optical/excitonic properties within the GW approximation with a priori inclusion of spin-orbit coupling and by solving the two-particle Bethe-Salpeter equation. Our results show that for all cases, the most stable stacking order is the high symmetry AA' type, distinctive of the bulk like 2H symmetry, followed by the AB stacking fault, typical of the 3R polytypism, which is by only 5 meV/formula unit less stable. The conduction band minimum is always located in the midpoint between K and Gamma, regardless of the stacking and chemical composition. All MX2 undergo an direct-to-indirect optical gap transition going from the monolayer to the bilayer regime. The stacking and the characteristic vdW interlayer distance mainly influence the valence band splitting at K and its relative energy with respect to Gamma, as well as, the electron-hole binding energy and the values of the optical excitations.mixedHe JG; Hummer K; Franchini CHe JG; Hummer K; Franchini

Effect of spin-orbit interaction on the optical spectra of single-layer, double-layer, and bulk MoS2

We present converged ab initio calculations of the optical absorption spectra of single-layer, double-layer, and bulk MoS2. Both the quasiparticle-energy calculations (on the level of the GW approximation ) and the calculation of the absorption spectra (on the level of the Bethe-Salpeter equation) explicitly include spin-orbit coupling, using the full spinorial Kohn-Sham wave functions as input. Without excitonic effects, the absorption spectra would have the form of a step function, corresponding to the joint density of states of a parabolic band dispersion in two dimensions. This profile is deformed by a pronounced bound excitonic peak below the continuum onset. The peak is split by spin-orbit interaction in the case of single-layer and (mostly) by interlayer interaction in the case of double-layer and bulk MoS2. The resulting absorption spectra are thus very similar in the three cases, but the interpretation of the spectra is different. Differences in the spectra can be seen in the shape of the absorption spectra at 3 eV where the spectra of the single and double layers are dominated by a strongly bound exciton

Photoemission of Bi₂Se₃ with Circularly Polarized Light: Probe of Spin Polarization or Means for Spin Manipulation?

Topological insulators are characterized by Dirac-cone surface states with electron spins locked perpendicular to their linear momenta. Recent theoretical and experimental work implied that this specific spin texture should enable control of photoelectron spins by circularly polarized light. However, these reports questioned the so far accepted interpretation of spin-resolved photoelectron spectroscopy.We solve this puzzle and show that vacuum ultraviolet photons (50–70 eV) with linear or circular polarization indeed probe the initial-state spin texture of Bi₂Se₃ while circularly polarized 6-eV low-energy photons flip the electron spins out of plane and reverse their spin polarization, with its sign determined by the light helicity. Our photoemission calculations, taking into account the interplay between the varying probing depth, dipole-selection rules, and spin-dependent scattering effects involving initial and final states, explain these findings and reveal proper conditions for light-induced spin manipulation. Our results pave the way for future applications of topological insulators in optospintronic devices

Soft Law as a New Mode of Governance: A Legal Perspective

After a brief review of the history and typology of soft law in public international law, we approach the concept deductively. We reject the binary view and subscribe to the continuum view. Building on the idea of graduated normativity and on the prototype theory of concepts, we submit that soft law is in the penumbra of law. It can be distinguished from purely political documents more or less readily, depending on its closeness to the prototype of law. Insights gained by the study of public international soft law are relevant to EC and EU soft law despite some differences between those legal orders. European soft law is created by institutions, Member States, and private actors. The legal effects of soft law acts can be clustered according to their relation to hard law. Both practical and normative considerations motivate reliance on soft law. An examination of the soft legal consequences of a disregard of soft law shows that compliance control mechanisms for hard and soft international law are converging. Moreover, some factors of compliance are independent of the theoretical hardness or softness of a given norm. In a legal policy perspective, the proliferation of soft law carries both dangers and benefits. Especially soft acts with a lawplus function do not weaken the respective regimes, but perfect them