Raman-scattering study of the phonon dispersion in twisted bi-layer graphene

Bi-layer graphene with a twist angle \theta\ between the layers generates a superlattice structure known as Moir\'{e} pattern. This superlattice provides a \theta-dependent q wavevector that activates phonons in the interior of the Brillouin zone. Here we show that this superlattice-induced Raman scattering can be used to probe the phonon dispersion in twisted bi-layer graphene (tBLG). The effect reported here is different from the broadly studied double-resonance in graphene-related materials in many aspects, and despite the absence of stacking order in tBLG, layer breathing vibrations (namely ZO' phonons) are observed.Comment: 18 pages, 4 figures, research articl

Group Theory analysis of phonons in two-dimensional Transition Metal Dichalcogenides

Transition metal dichalcogenides (TMDCs) have emerged as a new two dimensional materials field since the monolayer and few-layer limits show different properties when compared to each other and to their respective bulk materials. For example, in some cases when the bulk material is exfoliated down to a monolayer, an indirect-to-direct band gap in the visible range is observed. The number of layers $N$ ($N$ even or odd) drives changes in space group symmetry that are reflected in the optical properties. The understanding of the space group symmetry as a function of the number of layers is therefore important for the correct interpretation of the experimental data. Here we present a thorough group theory study of the symmetry aspects relevant to optical and spectroscopic analysis, for the most common polytypes of TMDCs, i.e. $2Ha$, $2Hc$ and $1T$, as a function of the number of layers. Real space symmetries, the group of the wave vectors, the relevance of inversion symmetry, irreducible representations of the vibrational modes, optical selection rules and Raman tensors are discussed.Comment: 32 pages, 4 figure

Group theory analysis of electrons and phonons in N-layer graphene systems

In this work we study the symmetry properties of electrons and phonons in graphene systems as function of the number of layers. We derive the selection rules for the electron-radiation and for the electron-phonon interactions at all points in the Brillouin zone. By considering these selection rules, we address the double resonance Raman scattering process. The monolayer and bilayer graphene in the presence of an applied electric field are also discussed.Comment: 8 pages, 6 figure

Nature of the constant factor in the relation between radial breathing mode frequency and tube diameter for single-wall carbon nanotubes

Resonance Raman scattering is used to determine the radial breathing mode (RBM) frequency (ωRBM) dependence on tube diameter (dt) for single-wall carbon nanotubes (SWNTs). We establish experimentally the ωRBM=227.0/dt as the fundamental relation for pristine SWNTs. All the other RBM values found in the literature can be explained by an upshift in frequency due mostly to van der Waals interaction between SWNTs and environment

How does the substrate affect the Raman and excited state spectra of a carbon nanotube?

We study the optical properties of a single, semiconducting single-walled carbon nanotube (CNT) that is partially suspended across a trench and partially supported by a SiO2-substrate. By tuning the laser excitation energy across the E33 excitonic resonance of the suspended CNT segment, the scattering intensities of the principal Raman transitions, the radial breathing mode (RBM), the G-mode and the D-mode show strong resonance enhancement of up to three orders of magnitude. In the supported part of the CNT, despite a loss of Raman scattering intensity of up to two orders of magnitude, we recover the E33 excitonic resonance suffering a substrate-induced red shift of 50 meV. The peak intensity ratio between G-band and D-band is highly sensitive to the presence of the substrate and varies by one order of magnitude, demonstrating the much higher defect density in the supported CNT segments. By comparing the E33 resonance spectra measured by Raman excitation spectroscopy and photoluminescence (PL) excitation spectroscopy in the suspended CNT segment, we observe that the peak energy in the PL excitation spectrum is red-shifted by 40 meV. This shift is associated with the energy difference between the localized exciton dominating the PL excitation spectrum and the free exciton giving rise to the Raman excitation spectrum. High-resolution Raman spectra reveal substrate-induced symmetry breaking, as evidenced by the appearance of additional peaks in the strongly broadened Raman G band. Laser-induced line shifts of RBM and G band measured on the suspended CNT segment are both linear as a function of the laser excitation power. Stokes/anti-Stokes measurements, however, reveal an increase of the G phonon population while the RBM phonon population is rather independent of the laser excitation power.Comment: Revised manuscript, 20 pages, 8 figure

Fermionic current densities induced by magnetic flux in a conical space with a circular boundary

We investigate the vacuum expectation value of the fermionic current induced by a magnetic flux in a (2+1)-dimensional conical spacetime in the presence of a circular boundary. On the boundary the fermionic field obeys MIT bag boundary condition. For irregular modes, a special case of boundary conditions at the cone apex is considered, when the MIT bag boundary condition is imposed at a finite radius, which is then taken to zero. We observe that the vacuum expectation values for both charge density and azimuthal current are periodic functions of the magnetic flux with the period equal to the flux quantum whereas the expectation value of the radial component vanishes. For both exterior and interior regions, the expectation values of the current are decomposed into boundary-free and boundary-induced parts. For a massless field the boundary-free part in the vacuum expectation value of the charge density vanishes, whereas the presence of the boundary induces nonzero charge density. Two integral representations are given for the boundary-free part in the case of a massive fermionic field for arbitrary values of the opening angle of the cone and magnetic flux. The behavior of the induced fermionic current is investigated in various asymptotic regions of the parameters. At distances from the boundary larger than the Compton wavelength of the fermion particle, the vacuum expectation values decay exponentially with the decay rate depending on the opening angle of the cone. We make a comparison with the results already known from the literature for some particular cases.Comment: 34 pages, 6 figure

Thermodynamics of the incommensurate state in Rb_2WO_4: on the Lifshitz point in A`A``BX_4 compounds

We consider the evolution of the phase transition from the parent hexagonal phase $P6_{3}/mmc$ to the orthorhombic phase $Pmcn$ that occurs in several compounds of $A'A''BX_{4}$ family as a function of the hcp lattice parameter $c/a$. For compounds of $K_{2}SO_{4}$ type with $c/a$ larger than the threshold value 1.26 the direct first-order transition $Pmcn-P6_{3}/mmc$ is characterized by the large entropy jump $Rln2$. For compounds $Rb_{2}WO_{4}$, $K_{2}MoO_{4}$, $K_{2}WO_{4}$ with $c/a<1.26$ this transition occurs via an intermediate incommensurate $(Inc)$ phase. DSC measurements were performed in $Rb_{2}WO_{4}$ to characterize the thermodynamics of the $Pmcn-Inc-P6_{3}/mmc$ transitions. It was found that both transitions are again of the first order with entropy jumps $0.2Rln2 and$0.3Rln2$. Therefore, at$c/a ~ 1.26$the$A'A''BX_{4}$compounds reveal an unusual Lifshitz point where three first order transition lines meet. We propose the coupling of crystal elasticity with$BX_{4}$ tetrahedra orientation as a possible source of the transitions discontinuity.Comment: 13 pages,1 Postscript figure. Submitted as Brief Report to Phys. Rev. B, this paper reports a new work in Theory and Experiment, directed to Structural Phase Transition

Basal-plane Incommensurate Phases in HCP Structures

An Ising model with competing interaction is used to study the appearance of incommensurate phases in the basal plane of an hexagonal closed-packed structure. The calculated mean-field phase diagram reveals various 1q-incommensurate and lock-in phases. The results are applied to explain the basal-plane incommensurate phase in some compounds of the A'A"BX_4 family, like K_2MoO_4, K_2WO_4, Rb_2WO4 and to describe the sequence of high-temperature phase transitions in other compounds of this family.Comment: 8 pages, RevTeX + 4 ps figure