Longitudinal Optical Modes Research Articles

Electron-phonon coupling and superconductivity in the doped topological crystalline insulator (Pb0.5Sn0.5)1−xInxTe

We present a neutron scattering study of phonons in single crystals of (Pb$_{0.5}$Sn$_{0.5}$)$_{1-x}$In$_x$Te with $x=0$ (metallic, but nonsuperconducting) and $x=0.2$ (nonmetallic normal state, but superconducting). We map the phonon dispersions (more completely for $x=0$) and find general consistency with theoretical calculations, except for the transverse and longitudinal optical (TO and LO) modes at the Brillouin zone center. At low temperature, both modes are strongly damped but sit at a finite energy ($\sim4$ meV in both samples), shifting to higher energy at room temperature. These modes are soft due to a proximate structural instability driven by the sensitivity of Pb-Te and Sn-Te $p$-orbital hybridization to off-center displacements of the metal atoms. The impact of the soft optical modes on the low-energy acoustic modes is inferred from the low thermal conductivity, especially at low temperature. Given that the strongest electron-phonon coupling is predicted for the LO mode, which should be similar for both studied compositions, it is intriguing that only the In-doped crystal is superconducting. In addition, we observe elastic diffuse (Huang) scattering that is qualitatively explained by the difference in Pb-Te and Sn-Te bond lengths within the lattice of randomly distributed Pb and Sn sites. We also confirm the presence of anomalous diffuse low-energy atomic vibrations that we speculatively attribute to local fluctuations of individual Pb atoms between off-center sites.

Temperature-Dependent Cationic Doping-Driven Phonon Dynamics Investigation in CdO Thin Films Using Raman Spectroscopy

In the present work, undoped cadmium oxide (CdO)- and 1% Tin (Sn)-doped CdO thin films were prepared by the sol–gel route. These samples have been analyzed by temperature (T)-dependent (80–500 K) Raman spectroscopy and studied for their lattice dynamics and vibrational density of states. Results indicate that the room T synthesized pure CdO thin film manifests two prime second order features, that is, 300.5 cm–1 transverse optical (TO) and 488 cm–1 longitudinal optical (LO) phonon modes. However, incorporation of cationic-assisted impurity (Sn) with larger ionic radii results in a softening of the high-frequency LO (480.5 cm–1) mode via lattice deformation scattering potential. In fact, selective Sn doping increases the carrier concentration in the host CdO matrix which subsequently mitigates intraionic anharmonicity owing to increased Coulomb screening, leading to disappearance of the low-frequency TO mode (300.5 cm–1). In the case of pure CdO thin films, surface electron-induced electron–LO phonon coupling causes the intensity enhancement in LO modes, while negligible four-phonon anharmonic coupling results in lowering of full width at half maxima below Debye T. On the other hand, Frühlich interaction in the polar LO phonon mode supersedes the impact of anharmonic decay and dominates the overall phonon decay process by impurity incorporation, via appropriate Sn doping. Theoretical phonon dispersion profiles throughout the Brillouin zone with increasing T suggests stronger TO phonon mode softening along with optical branch broadening followed by LO–TO splitting. The acoustic branch barely suffers any shift with changing T which cannot be observed in experimental spectra. However, the flexural phonon modes confer a direct indication of changed rigidity and bond stiffness with varying T. Overall, the present investigation provides experimental evidence regarding the significance of Debye T, below and above which the three- and four-phonon anharmonicity are feasible in phonon scattering in conjunction with theoretical insights.

Investigation on ZnTe/CdxZn1-xTe heterostructure for photodetector applications

ZnTe/CdxZn1-xTe (0.2 ≤ x ≤ 1.0) heterostructures were fabricated by thermal evaporation method by using CdTe and ZnTe as source materials. The properties of heterostructures were analysed by X-ray diffraction (XRD), photoluminescence (PL), temperature dependent Raman spectroscopy, X-ray photoelectron spectroscopy (XPS) and I-V characterizations. Double diffraction peak in the diffractogram has confirmed bilayer with no or minimum diffusion across the junction. Effect of thermal expansion and an-harmonic coupling to the other phonon modes causing strain induced shift of the longitudinal optical modes were studied through low-temperature Raman spectra. The junction parameters such as barrier height, ideality factor and series resistance were extracted from I - V characteristics by applying various models. ZnTe/CdxZn1-xTe heterostructures having low barrier height and good photo response are explored in this work for the photodetector application. Sample having composition x = 0.8 showed higher response in the visible region with improved response at longer wavelengths.

Anisotropic interlayer exciton in black phosphorus van der Waals heterostructures

We study theoretically the variations of interlayer exciton binding energy caused by the exciton–optical phonons coupling in van der Waals heterostructures composed of double-layer black phosphorus (BP), where both the longitudinal optical phonon modes and the surface optical phonon modes induced by the inserted h-BN layer are taken into account. We find that the strength of exciton–phonon coupling depends on the anisotropy of the effective mass of interlayer exciton, which plays a crucial rule in adjusting the binding energies of interlayer excitons. Moreover, the modulation effects of interlayer distance between two layers as well as the internal distance between BP layer and the inserted h-BN layer on the binding energies are analyzed quantitatively. These results provide theoretical insight when modulating interlayer excitons in two-dimensional van der Waals heterostructures in experiments.

Ultrafast dynamics of hot carriers in a quasi–two-dimensional electron gas on InSe

Two-dimensional electron gases (2DEGs) are at the base of current nanoelectronics because of their exceptional mobilities. Often the accumulation layer forms at polar interfaces with longitudinal optical (LO) modes. In most cases, the many-body screening of the quasi-2DEGs dramatically reduces the Fröhlich scattering strength. Despite the effectiveness of such a process, it has been recurrently proposed that a remote coupling with LO phonons persists even at high carrier concentration. We address this issue by perturbing electrons in an accumulation layer via an ultrafast laser pulse and monitoring their relaxation via time- and momentum-resolved spectroscopy. The cooling rate of excited carriers is monitored at doping level spanning from the semiconducting to the metallic limit. We observe that screening of LO phonons is not as efficient as it would be in a strictly 2D system. The large discrepancy is due to the remote coupling of confined states with the bulk. Our data indicate that the effect of such a remote coupling can be mimicked by a 3D Fröhlich interaction with Thomas-Fermi screening. These conclusions are very general and should apply to field effect transistors (FET) with high-κ dielectric gates, van der Waals heterostructures, and metallic interfaces between insulating oxides.

Phononic and electronic features of diamond nanowires

By using density functional theory, we have studied vibrational features, Raman activities, and electronic properties of ultrathin hydrogen-passivated diamond nanowires (H-DNWs). Confinement imposes the softening of transverse acoustic modes and the hardening of the low-optical modes. Higher frequencies optical modes are more influenced by the surface termination effect. Surface effect splits transverse (TO) and longitudinal (LO) optical modes with a redshift from the bulk diamond. Electronic inspection shows that the surface terminated hydrogen atoms have the main contribution to the conduction bands of H-DNWs. This causes, contrary to our expectations from the quantum confinement effect, the electronic bandgap of H-DNWs becomes smaller than bulk diamond.

Multipolar scattering of subwavelength interacting particles: Extraction of effective properties between transverse and longitudinal optical modes

The general concern of this investigation is the extraction of the effective electromagnetic properties of agglomerates of randomly located particles, which are small compared to the wavelength. The focus is on the spectral window ranging from the transverse phonon to the longitudinal phonon frequencies, in which resonances may be excited. In this frequency domain, it is shown that limiting the problem to bare dipole-dipole interactions leads to inaccurate calculations of the electromagnetic fields, resulting in a doubtful extraction of the effective properties. We perform the complete electromagnetic calculation by taking into account the higher multipole orders that are activated when the agglomerates are illuminated. Several results, which are not usually observed outside the frequency range highlighted here, are revealed by means of extensive numerical simulations. In particular, we evidence large deviations compared to the predictions provided by the effective medium theories, while we find that the volume of the microstructures (each one with a different internal geometry) that are used to average the fields must be unusually large to avoid a bias in the determination of the effective properties. Furthermore, the evolution of the incoherent component of the fields between the two optical phonon modes is investigated.

Contrasting temperature dependence of the band gap in CH3NH3PbX3 (X=I, Br, Cl) : Insight from lattice dilation and electron-phonon coupling

Although hybrid halide perovskites $(\text{MAPb}{X}_{3},$ $\mathrm{MA}={\mathrm{CH}}_{3}{\mathrm{NH}}_{3} \mathrm{and} X=\mathrm{I}, \mathrm{Br}, \mathrm{Cl})$ have been ubiquitously explored from the photovoltaic perspective, there are still a few unanswered questions which require a more fundamental understanding. One such unsettled issue is the puzzling behavior of the band gap. Unlike conventional semiconductors, $\text{MAPb}{X}_{3}$ $(X=\mathrm{I}, \mathrm{Br})$ is found to show a blueshift (increase) in the band gap $({E}_{g})$ with increasing temperature $(T)$, while ${\mathrm{MAPbCl}}_{3}$ shows an unusual redshift (decrease). In order to understand this, we performed a detailed $T$-dependent study of electronic, optical, and structural properties of $\text{MAPb}{X}_{3}$ combined with the state-of-the-art first-principles calculations. With increasing $T$, two dominant mechanisms which come into play are lattice dilation and electron-phonon coupling (EPC). The former (latter) is responsible for an increase (decrease) in ${E}_{g}$. We found that lattice dilation effect dominates in $\text{MAPb}{X}_{3}$ $(X=\mathrm{I}, \mathrm{Br})$, causing an enhancement in ${E}_{g}$. EPC involves various contributions, of which the interaction of charge carriers with the longitudinal optical phonon mode via Fr\ohlich interaction is the most dominant one at room temperature. We quantify this contribution using Fr\ohlich's theory of large polarons and show that the ${E}_{g}$ correction due to this effect in ${\mathrm{MAPbCl}}_{3}$ is almost double as compared to ${\mathrm{MAPbBr}}_{3}$ and thus explain the reduction in ${E}_{g}$ for the former.

Brillouin zone center phonon modes in ZnGa2O4

Infrared-active lattice mode properties of melt-grown high-quality single bulk crystals of ZnGa2O4 are investigated by combined spectroscopic ellipsometry and density functional theory computation analysis. The normal spinel structure crystals are measured by spectroscopic ellipsometry at room temperature in the range of 100 cm–1–1200 cm–1. The complex-valued dielectric function is determined from a wavenumber-by-wavenumber approach, which is then analyzed by the four-parameter semi-quantum model dielectric function approach augmented by impurity mode contributions. We determine four infrared-active transverse and longitudinal optical mode pairs, five localized impurity mode pairs, and the high frequency dielectric constant. All four infrared-active transverse and longitudinal optical mode pairs are in excellent agreement with results from our density functional theory computations. With the Lyddane–Sachs–Teller relationship, we determine the static dielectric constant, which agrees well with electrical capacitance measurements performed on similarly grown samples. We also provide calculated parameters for all Raman-active and for all silent modes and, thereby, provide a complete set of all symmetry predicted Brillouin zone center modes.

MORPHOLOGICAL, STRUCTURAL AND OPTICAL CHARACTERIZATIONS OF Zn-DOPED CdS BUFFER LAYER ELABORATED BY CHEMICAL BATH DEPOSITION

Zn-doped CdS layers, with various percentage ratios [Formula: see text] (= [Zn[Formula: see text]]/[Cd[Formula: see text]]%) were grown on glass substrates by chemical bath deposition (CBD). The effect of Zn-doping on different properties of CdS is studied by X-ray diffraction (XRD), Raman spectroscopy, scanning electron microscopy, atomic force microscopy, energy dispersive X-ray analysis and UV-visible diffuse reflectance. The XRD patterns indicated polycrystalline films with (111) orientation and the insertion of Zn does not change the crystallinity of CdS. The Raman spectra show one major peak centered around 300[Formula: see text]cm[Formula: see text] assigned to the first-order longitudinal optic (LO) phonon modes of CdS. The surface morphology visualized by AFM and SEM analysis showed the influence of the Zn-doping on the morphology of the films, the surface roughness is found to decrease from 16.5 to 8.9[Formula: see text]nm with augmenting the ratio [Formula: see text] from 0 to 6%. In regard to the SEM analysis, the increase of [Formula: see text] yielded a lower porosity of the film and voids, and the films become more homogeneous. The EDAX spectra confirmed the existence of Zn in the doped samples. The bandgap decreases from 2.44 to 2.37[Formula: see text]eV, while the transmittance increases from 76 to 86% with augmenting the ratio [Formula: see text].

Decoupling the effects of composition and strain on the vibrational modes of GeSn semiconductors

We report on the behavior of Ge-Ge, Ge-Sn, Sn-Sn like, and disorder-activated (DA) vibrational modes in GeSn semiconductors investigated using Raman scattering spectroscopy. By using an excitation wavelength close to the E1 gap, all modes are clearly resolved and their evolution as a function of strain and Sn content is established. Previous Raman scattering studies mainly focused on the Ge-Ge peak position which is insufficient to evaluate the effects of lattice strain and Sn content. Herein to decouple the individual contributions of content and strain, the analysis was conducted on a series of pseudomorphic and relaxed epitaxial layers with a Sn content in the 5–17 at.% range. The frequencies of all vibrational modes were found to display qualitatively the same behavior as a function of content and strain, that is a linear downshift as the Sn content increases or the compressive strain relaxes. Simultaneously, the Ge-Sn and Ge-Ge peaks broaden, and the latter becomes increasingly asymmetric. The behavior of the integrated intensity, width, and asymmetry of each one of these vibrational modes was also evaluated. We found that an increase in Sn content is associated with an increase in the relative integrated intensity of Ge-Sn and DA modes. The latter also increases as the layers become more compressively strained and become more prominent under the configuration as the intensity of the adjacent longitudinal optical modes decreases. The Raman mode asymmetry, coupled with the peak position, is exploited to implement an empirical approach to accurately quantify the Sn composition and lattice strain from Raman spectra.

Nondestructive Characterizations of Au-Catalyzed GaAs Nanowires on GaAs(111)B Substrates via Identifications of 1st Order Optical Phonon Modes Using μ-Raman Spectroscopy.

We report the relation between the catalyst patterning conditions and the intensity of the 1st order Raman active modes in Au-catalyzed GaAs nanowire bundles. We fabricated e-beam lithographically Au-patterned GaAs(111)B substrates by varying the patterning conditions (e-beam dose rate, dot-size and interdot-spacings), and grew GaAs nanowires via vapor-liquid-solid process using a solid-source molecular beam epitaxy. To understand the effects of the substrate preparation conditions and resulting morphologies on the optical characteristics of 1st order transverse optical and longitudinal optical phonon modes of GaAs, we characterized the nanowire bundles using complementary μ-Raman spectroscopy and scanning electron microscopy as a function of the e-beam dose rate (145-595 μC/cm²), inter-dot spacing (100 and 150 nm) and pattern size (100 and 150 nm). Ensembles of single crystalline GaAs nanowires covered with different Au-thickness exhibit a downshift and asymmetric broadening of the 1st order transverse optical and longitudinal optical phonon peaks relative to GaAs bulk modes. We also showed that the sensitivity of a downshift and broadening of Raman spectra are directly related to morphological and surface coverage variations in as-grown nanowires. We observed clear increases of the transverse optical and longitudinal optical intensity as well as the relatively higher peak shift and broadening of Raman spectra from the 100 nm patterning in response to the dose rate change. Strong dependence of Raman spectra of the nanowire bundles on the e-beam dose rate changes are attributed to the variations in spatial density, size, shape and random growth orientation of the wires. We have shown that the identification of the changes in GaAs longitudinal optical and Arsenic anti-site peaks is good indicators to characterize the quality of as-grown GaAs nanowires. Our finding confirms the utilization of Raman spectroscopy as a powerful tool for characterizing chemical, structural, and morphological information of as-grown nanowires within the supporting substrate.

Bandgap engineering of low-temperature CdS nanocrystalline prepared on Si(1 1 1) without post-thermal annealing

This article reports a new attempt for the bandgap formation of cadmium sulfide nanocrystals (CdS NCs) synthesized on Si(1 1 1) substrates at low temperatures: 50 and 150 °C. The influence of energy gap alteration on the optoelectronic properties was extensively considered with the aid of various tools. A wurtzite-–hexagonal structure in both CdS/Si NCs samples was confirmed by X-ray diffraction measurements. For a crystallite size around 7 nm, wider energy gaps (2.94–2.98 eV) were revealed by photoluminescence spectra. These values are in good conformity with the precise energy gaps, 2.95 eV and 3.04 eV which were estimated using a reflection-electron energy loss spectroscopy. Consequently, such expansion in the energy bandgap indicates the occurrence of a blue shift compared to the bulk CdS. Moreover, the Raman spectra show three peaks referring to the first-, second-, and third-order scattering of the longitudinal optical phonon modes. The findings demonstrate the improved crystalline quality of the sample prepared at 150 °C with no need for further thermal annealing. It is thus inferred that quantum confinement is possible in the CdS NCs prepared at low temperatures using a pulsed-laser deposition technique; thereby, they are useful for optoelectronic applications.

Phase-controlled metasurface design via optimized genetic algorithm

Abstract In an optical Pancharatnam-Berry (PB) phase metasurface, each sub-wavelength dielectric structure of varied spatial orientation can be treated as a point source with the same amplitude yet varied relative phase. In this work, we introduce an optimized genetic algorithm (GA) method for the synthesis of one-dimensional (1D) PB phase-controlled dielectric metasurfaces by seeking for optimized phase profile solutions, which differs from previously reported amplitude-controlled GA method only applicable to generate transverse optical modes with plasmonic metasurfaces. The GA–optimized phase profiles can be readily used to construct dielectric metasurfaces with improved functionalities. The loop of phase-controlled GA consists of initialization, random mutation, screened evolution, and duplication. Here random mutation is realized by changing the phase of each unit cell, and this process should be efficient to obtain enough mutations to drive the whole GA process under supervision of appropriate mutation boundary. A well-chosen fitness function ensures the right direction of screened evolution, and the duplication process guarantees an equilibrated number of generated light patterns. Importantly, we optimize the GA loop by introducing a multi-step hierarchical mutation process to break local optimum limits. We demonstrate the validity of our optimized GA method by generating longitudinal optical modes (i. e., non-diffractive light sheets) with 1D PB phase dielectric metasurfaces having non-analytical counter-intuitive phase profiles. The produced large-area, long-distance light sheets could be used for realizing high-speed, low-noise light-sheet microscopy. Additionally, a simplified 3D light pattern generated by a 2D PB phase metasurface further reveals the potential of our optimized GA method for manipulating truly 3D light fields.

Anomalous low energy phonon dispersion in bulk silicon-germanium observed by inelastic x-ray scattering

We report on an anomalous mode distinct from both optical and acoustic modes in phonon dispersion curves of bulk Si1−xGex alloy with x taking the values of 0.16, 0.32, 0.45, and 0.72. The anomalous mode at approximately 13 meV was observed directly using inelastic x-ray scattering along the Γ–X ([00q]) direction. The phonon dispersion relations of the anomalous mode indicate that there was no momentum dependence, similar to those of the longitudinal and transverse optical modes (Ge–Ge, Si–Ge, and Si–Si modes). In contrast to the acoustic and optical phonon modes, the energy of the anomalous mode shows no Ge fraction dependence. The molecular dynamics simulation corroborates that the Ge–Ge pairs or Ge atom clusters, which are surrounded by Si atoms, provide the anomalous mode, which is unique to the alloy structure. It has been suggested that such a localized vibration mode with no propagation significantly affects the acoustic modes, leading to low thermal conductivity in the SiGe alloy.

Oblique incidence infrared reflectance spectroscopy of phonons in cubic MgO, MnO, and NiO

The infrared (IR) reflectivity of the cubic metal oxides MgO, MnO, and NiO has been measured at room temperature using the technique of oblique incidence. The use of this technique at three angles of incidence provides multiple sets of spectra (including both s- and p-polarizations) to analyze when compared to the standard normal incidence case, which has been used extensively in the past for these compounds. It is shown that the transverse optic (TO) and longitudinal optic (LO) mode phonon parameters can be determined with greater accuracy by using the factorized model for the fits, as compared with the popular classical model used previously, and by fitting the derivative of the reflectivity. Our results for the phonon mode parameters are similar to those found earlier, as could be expected, but are generally more precise. An analysis of the difference in frequency of the TO mode in antiferromagnetic NiO at room temperature for the two polarizations of reflected light revealed a TO mode splitting of about 4 cm−1, with the p-polarized light TO mode having the higher frequency: This small splitting is in agreement with theoretical predictions. This more precise IR method for revealing the phonon mode behavior in such magnetically ordered cubic metal oxides, which are prototypes of strongly correlated electronic systems and have been found to be Mott-Hubbard insulators, may be readily applied to any similar antiferromagnetic system.

Optical, structural and morphological study of CdS nanoparticles: role of sulfur source

Cadmium sulfide (CdS) nanoparticles were synthesized by using a simple and low-cost home-made hot-injection method at a low process temperature using different sulfur (S) sources. The effects of sulfur concentration on the structural, morphological and optoelectronic properties of the synthesized cadmium sulfide films were studied using a range of characterization techniques: X-ray diffraction (XRD), Raman spectroscopy, field-emission scanning electron microscopy (FESEM) and ultraviolet–visible spectroscopy. The XRD studies revealed the formation of hexagonal-type cadmium sulfide nanoparticles. The varying morphology dependence on the sulfur source was ascertained from FESEM analysis. The longitudinal optical phonon vibrational modes of cadmium sulfide were assigned in Raman spectra at 300 and 600 cm−1. The bandgap of the cadmium sulfide particles was estimated to be 2·30 eV from Tauc’s plots. Consistent with the experimental results, this study’s first-principles density functional theory calculations predict the bandgap of cadmium sulfide nanoparticles to increase with decreasing sulfur concentration – Cd52S52 (2·38 eV), Cd52S51 (2·52 eV) and Cd52S50 (2·65 eV) – with both the valence and conduction band edges demonstrated to be dominated by sulfur p states.

Modeling of Electronic Mobilities in Halide Perovskites: Adiabatic Quantum Localization Scenario.

The transport properties of MAPbI3 are analyzed within a tight-binding model. We find a strong Fröhlich interaction of electron and holes with the electrostatic potential induced by the longitudinal optical phonon modes. This potential induces a strong scattering and limits the electronic mobilities at room temperature to about 200 cm^{2}/V s. With additional extrinsic disorder, a large fraction of the electrons and holes are localized, but they can diffuse by following nearly adiabatically the evolution of the electrostatic potential. This process of diffusion, at a rate which is given by the lattice dynamics, contributes to the unique electronic properties of this material.

Quantitative analysis of diffraction and infra-red spectra of composite cement/BaSO4/Fe3O4 for determining correlation between attenuation coefficient, structural and optical properties

To better understand the structural and optical properties of composite cement/BaSO4/Fe3O4 for various amount of BaSO4/Fe3O4, the X-ray diffraction (XRD) and Fourier transform infrared (FTIR) spectroscopy have been used to investigate the correlation between their structural and optical properties. The structural properties including crystallite size, micro strain, stress and energy deformation were analyzed from the quantitative analysis of XRD spectra by using size-strain plot (SSP) methods. The refractive index (n), extinction coefficient (k), dielectric functions (ε), and energy loss function (Im (−1/ε)) were analyzed from the quantitative analysis of FTIR spectra by using kramers-kronig (K–K) relations. The corresponding structures for high amount of BaSO4/Fe3O4 in composite cement/BaSO4/Fe3O4 become less stable which consistent with the distance between the wavenumber of transversal and longitudinal optical phonon vibration mode become shorter. For all composites cement/BaSO4/Fe3O4 in this study, we found that the distance wavenumber (Δ) between longitudinal optical (LO) and transversal optical (TO) phonon vibration decrease with increasing the crystallite size and linear attenuation coefficient. Our results indicated that the FTIR spectra could be useful for determining the optical phonon vibration, dielectric function, and energy loss function of composite cement/BaSO4/Fe3O4.

Revisiting longitudinal optical modes in materials with plasmon and plasmon-like absorptions – SrTiO3 and β-Ga2O3

We investigate exemplary the longitudinal optical (LO) mode order in compounds with a plasmon or plasmon-like phonon mode and additional phonon modes. When the oscillator strength of the plasmon or plasmon-like mode is gradually increased, a reordering of the modes takes place. Since it is not possible in crystals with orthorhombic or higher symmetry that a LO mode crosses a transverse optical (TO) mode's position, this reordering takes place via mode hybridization. During this mode hybridization, the plasmon or plasmon-like LO mode gradually becomes the originally higher situated LO mode while the latter morphs into the former. As a consequence, inner (LO-TO) and an outer (TO-LO) mode pairs are formed. These insights can be readily transferred to other semiconductors or many mode materials with reststrahlen bands, as long as their dielectric tensor can be (partly) diagonalized, and allow easily correct mode assignments.