Dielectric Function Tensor Research Articles

First-principles calculations to investigate structural, elastic and electronic properties of chloro-perovskite NaMgCl3

In this work, we have investigated the structural, electronic, and optical properties of the halide-perovskite NaMgCl3 by performing first-principles calculations using a mixed basis of the full-potential linearized augmented plane wave (FP-LAPW) approach under the generalized gradient approximation (GGA-PBEsol), a revised Perdew-Burke-Ernzerhof, and the modified Becke-Johnson (TB-mBJ) implemented on the Wien2k package. The obtained results show that NaMgCl3 is more stable in the nonmagnetic phase.The compound is found to be thermodynamically and elastically stable. It is found that there is a good agreement between the calculated lattice constants and the values published in previous studies. The complete set of first-order elastic constants tensor has been calculated, implying mechanical stability. The Young's modulus, shear modulus, Poisson's ratio,Pugh's ratio, and universal anisotropy index have been additionally calculated. The atomic bonds along both x and y axis are stronger than those along the z-axis. The Debye temperature is calculated using the average sound velocity. From electronic band structures, we have found that NaMgCl3 is an insulator with a direct wide band gap ΓV→ΓC. The electronic charge densities were plotted.Analysis of the charge density distribution map reveals that the bondings in NaMgCl3 are ionic. The calculated density of states (DOS) indicates that the valence band is mainly dominated by Cl− anions, the conduction band is mainly dominated by Mg2+ cations, whereas Na+ has a negligible contribution. The optical characteristics, notably the dielectric function tensor, refractive function, reflectivity, absorption coefficient, photo-conductivity and transmittance were plotted and discussed. The optical properties confirm that NaMgCl3 is optically anisotropic in 10–24 eV range .The low transmittance and high absorption coefficient in the far ultra-violet region, between 90 and 250 nm, make this compound a good candidate for use as an ultraviolet shield, in vacuum ultraviolet lenses and in thermoelectric promising features due to his direct wide band gap.Experimental and theoretical data are scarce for this compound in an ilmenite-type structure.

Anisotropic excitonic photocurrent in β−Ga2O3

Polarization-dependent photocurrent spectra are measured on a (001) $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ Schottky photodetector, where the linear polarization of light is rotated within the ab plane. Three spectral peaks at 4.92, 5.15, and 5.44 eV are observed that vary in intensity with the optical polarization direction. The peak transition energies are consistent with excitons previously reported in $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ due to interband transitions modified by the valence-band $p$-orbital anisotropy and the electron-hole Coulombic attraction. The measured polarization dependence of the photocurrent matches our predictions based on electromagnetic simulations of anisotropic absorption using the complex dielectric function tensor extracted from previous ellipsometry studies. These results illustrate the dominance of excitonic absorption and photocurrent in $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ both below and above the band gap, demonstrate a combined theoretical/experimental understanding of anisotropic photocarrier generation, and validate previous atomistic band-structure calculations in this low-symmetry ultrawide-band-gap semiconductor.

Anisotropic dielectric function, direction dependent bandgap energy, band order, and indirect to direct gap crossover in α-(AlxGa1−x)2O3 (≤x≤1)

Mueller matrix spectroscopic ellipsometry is applied to determine anisotropic optical properties for a set of single-crystal rhombohedral structure α-(AlxGa1−x)2O3 thin films (0 ≤ x ≤ 1). Samples are grown by plasma-assisted molecular beam epitaxy on m-plane sapphire. A critical-point model is used to render a spectroscopic model dielectric function tensor and to determine direct electronic band-to-band transition parameters, including the direction dependent two lowest-photon energy band-to-band transitions associated with the anisotropic bandgap. We obtain the composition dependence of the direction dependent two lowest band-to-band transitions with separate bandgap bowing parameters associated with the perpendicular (bEg,⊥ = 1.31 eV) and parallel (bEg,|| = 1.61 eV) electric field polarization to the lattice c direction. Our density functional theory calculations indicate a transition from indirect to direct characteristics between α-Ga2O3 and α-Al2O3, respectively, and we identify a switch in band order where the lowest band-to-band transition occurs with polarization perpendicular to c in α-Ga2O3 whereas for α-Al2O3 the lowest transition occurs with polarization parallel to c. We estimate that the change in band order occurs at approximately 40% Al content. Additionally, the characteristic of the lowest energy critical point transition for polarization parallel to c changes from M1 type in α-Ga2O3 to M0 type van Hove singularity in α-Al2O3.

Optical and magneto-optical properties of ferromagnetic monolayer CrBr3 : A first-principles GW and GW plus Bethe-Salpeter equation study

The discovery of atomically thin two-dimensional (2D) magnetic semiconductors has triggered enormous research interest recently. In this work, we use first-principles many-body perturbation theory to study a prototypical 2D ferromagnetic semiconductor, monolayer chromium tribromide (CrBr$_3$). With broken time-reversal symmetry, spin-orbit coupling, and excitonic effects included through the full-spinor $GW$ and $GW$ plus Bethe-Salpeter equation ($GW$-BSE) methods, we compute the frequency-dependent layer polarizability tensor and dielectric function tensor that govern the optical and magneto-optical properties. In addition, we provide a detailed theoretical formalism for simulating magnetic circular dichroism, magneto-optical Kerr effect, and Faraday effect, demonstrating the approach with monolayer CrBr$_3$. Due to reduced dielectric screening in 2D and the localized nature of the Cr $d$ orbitals, we find strong self-energy effects on the quasiparticle band structure of monolayer CrBr$_3$ that give a 3.8 eV indirect band gap. Also, excitonic effects dominate the low-energy optical and magneto-optical responses in monolayer CrBr$_3$ where a large exciton binding energy of 2.3 eV is found for the lowest bright exciton state with excitation energy at 1.5 eV. We further find that the magneto-optical signals demonstrate strong dependence on the excitation frequency and substrate refractive index. Our theoretical framework for modeling optical and magneto-optical effects could serve as a powerful theoretical tool for future study of optoelectronic and spintronics devices consisting of van der Waals 2D magnets.

Understanding longitudinal optical oscillator strengths and mode order

A classical way of describing dielectric functions employs sums of contributions from damped harmonic oscillators. We derive the corresponding dispersion relations for the dielectric function and the inverse dielectric function to investigate and establish the relationship between the transversal and longitudinal optical oscillator strengths, which can differ, according to experimental results, by up to three orders of magnitude. We demonstrate that transversal and longitudinal oscillator strengths should be identical for a single oscillator and that the experimental differences are in this case due to the introduction of a dielectric background in the dispersion formula. For this effect we derive an exact correction. Based on this correction we further derive a modified Kramers-Kronig sum rule for the isotropic case as well as for the components of the inverse dielectric function tensor. For systems with more than one oscillator, our model provides oscillator strengths and LO resonance wavenumbers for uncoupled LO modes.

Infrared active phonons in monoclinic lutetium oxyorthosilicate

A combined generalized spectroscopic ellipsometry measurement and density functional theory calculation analysis is performed to obtain the complete set of infrared active phonon modes in Lu2SiO5 with a monoclinic crystal structure. Two different crystals, each cut perpendicular to a different crystal axis, are investigated. Ellipsometry measurements from 40to1200cm−1 are used to determine the frequency dependent dielectric function tensor elements. The eigendielectric displacement vector summation approach and the eigendielectric displacement loss vector summation approach, both augmented with anharmonic lattice broadening parameters proposed recently for low-symmetry crystal structures [Mock et al., Phys. Rev. B 95, 165202 (2017)], are applied for our ellipsometry data analysis. All measured and model calculated dielectric function tensor and inverse dielectric function tensor elements match excellently. 23 Au symmetry and 22 Bu symmetry infrared active transverse and longitudinal optical modes are found. We also determine the directional limiting modes and the order of the phonon modes within the monoclinic plane. Results from density functional theory and ellipsometry measurements are compared and nearly perfect agreement is observed. We further compare our results to those obtained recently for the monoclinic crystal Y2SiO5, which is isostructural to Lu2SiO5 [Mock et al., Phys. Rev. B 97, 165203 (2018)]. We find that the lattice mode behavior of monoclinic Lu2SiO5 is qualitatively identical with Y2SiO5 and differs only quantitatively. We anticipate that members of the isostructural group of monoclinic symmetry oxyorthosilicates such as Dy2SiO5 or Yb2SiO5 will likely behave very similar in their phonon mode properties as reported here for Lu2SiO5.

Mid-IR and UV-Vis-NIR Mueller matrix ellipsometry characterization of tunable hyperbolic metamaterials based on self-assembled carbon nanotubes

Mueller matrix ellipsometry over the wide spectral range from the mid-IR to UV is applied to characterize the dielectric function tensor for films of densely packed single-walled carbon nanotubes aligned in the surface plane. These films optically act as metamaterials with an in-plane anisotropic, bulk effective medium response. A parameterized oscillator model is developed to describe electronic interband transitions, π−π⋆ plasmon resonances, and free-carrier absorption. Wide ranges of hyperbolic dispersion are observed and exceptional tuneability of the hyperbolic ranges is demonstrated by comparing results for a film of aligned but unordered carbon nanotubes with a film fabricated under optimized conditions to achieve hexagonally close-packed alignment of the nanotubes. The effect of doping on the optical properties and hyperbolic range is discussed.

Temperature dependence of dielectric function spectra and interband optical transitions in layered TlInS2

The temperature behavior of the dielectric function spectra and related interband optical transitions in layered TlInS2 with a structural ferroelectric phase transition via an intermediate incommensurate phase has been studied over the temperature range of 80–400 K by using spectroscopic ellipsometry in the photon energy region from 1.5 to 4.2 eV. Both (100) and (001) surfaces of TlInS2 have been examined, and the collected ellipsometric data have allowed retrieving diagonal components of the dielectric function tensor. Standard critical point analysis of the retrieved components shows that energy and broadening of the interband optical transitions subsequently experience remarkable changes upon reaching the points of phase transitions. The obtained results provide a better insight into optical properties and the electronic structure of incommensurate semiconductors-ferroelectrics.

Anisotropy and phonon modes from analysis of the dielectric function tensor and the inverse dielectric function tensor of monoclinic yttrium orthosilicate

We determine the frequency dependence of the four independent Cartesian tensor elements of the dielectric function for monoclinic symmetry Y$_2$SiO$_5$ using generalized spectroscopic ellipsometry from 40-1200 cm$^{-1}$. Three different crystal cuts, each perpendicular to a principle axis, are investigated. We apply our recently described augmentation of lattice anharmonicity onto the eigendielectric displacement vector summation approach [A. Mock et al., Phys. Rev. B 95, 165202 (2017)], and we present and demonstrate the application of an eigendielectric displacement loss vector summation approach with anharmonic broadening. We obtain excellent match between all measured and model calculated dielectric function tensor elements and all dielectric loss function tensor elements. We obtain 23 A$_{\mathrm{u}}$ and 22 B$_{\mathrm{u}}$ symmetry long wavelength active transverse and longitudinal optical mode parameters including their eigenvector orientation within the monoclinic lattice. We perform density functional theory calculations and obtain 23 A$_{\mathrm{u}}$ symmetry and 22 B$_{\mathrm{u}}$ transverse and longitudinal optical mode parameters and their orientation within the monoclincic lattice. We compare our results from ellipsometry and density functional theory and find excellent agreement. We also determine the static and above reststrahlen spectral range dielectric tensor values and find a recently derived generalization of the Lyddane-Sachs-Teller relation for polar phonons in monoclinic symmetry materials satisfied [M. Schubert, Phys. Rev. Lett. 117, 215502 (2016)].

Band-to-band transitions, selection rules, effective mass, and excitonic contributions in monoclinic β−Ga2O3

We employ an eigenpolarization model including the description of direction dependent excitonic effects for rendering critical point structures within the dielectric function tensor of monoclinic $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ yielding a comprehensive analysis of generalized ellipsometry data obtained from 0.75--9 eV. The eigenpolarization model permits complete description of the dielectric response. We obtain, for single-electron and excitonic band-to-band transitions, anisotropic critical point model parameters including their polarization vectors within the monoclinic lattice. We compare our experimental analysis with results from density functional theory calculations performed using the Gaussian-attenuation-Perdew-Burke-Ernzerhof hybrid density functional. We present and discuss the order of the fundamental direct band-to-band transitions and their polarization selection rules, the electron and hole effective mass parameters for the three lowest band-to-band transitions, and their excitonic contributions. We find that the effective masses for holes are highly anisotropic and correlate with the selection rules for the fundamental band-to-band transitions. The observed transitions are polarized close to the direction of the lowest hole effective mass for the valence band participating in the transition.

Anisotropy, phonon modes, and lattice anharmonicity from dielectric function tensor analysis of monoclinic cadmium tungstate

We determine the frequency dependence of four independent CdWO$_4$ Cartesian dielectric function tensor elements by generalized spectroscopic ellipsometry within mid-infrared and far-infrared spectral regions. Single crystal surfaces cut under different angles from a bulk crystal, (010) and (001), are investigated. From the spectral dependencies of the dielectric function tensor and its inverse we determine all long wavelength active transverse and longitudinal optic phonon modes with $A_u$ and $B_u$ symmetry as well as their eigenvectors within the monoclinic lattice. We thereby demonstrate that such information can be obtained completely without physical model line shape analysis in materials with monoclinic symmetry. We then augment the effect of lattice anharmonicity onto our recently described dielectric function tensor model approach for materials with monoclinic and triclinic crystal symmetries [Phys. Rev. B, 125209 (2016)], and we obtain excellent match between all measured and modeled dielectric function tensor elements. All phonon mode frequency and broadening parameters are determined in our model approach. We also perform density functional theory phonon mode calculations, and we compare our results obtained from theory, from direct dielectric function tensor analysis, and from model lineshape analysis, and we find excellent agreement between all approaches. We also discuss and present static and above reststrahlen spectral range dielectric constants. Our data for CdWO$_4$ are in excellent agreement with a recently proposed generalization of the Lyddane-Sachs-Teller relation for materials with low crystal symmetry [Phys. Rev. Lett. 117, 215502 (2016)].

Optical properties of lithium gallium oxide

The optical dielectric function tensor of orthorhombic single-crystal LiGaO2 was determined for polarizations along a, b, and c crystal-axis in the photon energy range from 0.04eV to 6.5eV by the generalized spectroscopic ellipsometry. In the far-infrared spectral range from 12.4meV to 40meV, the dielectric function was determined from conventional polarized transmittance and polarized reflectance measurements. Lineshape analysis of the dielectric function tensor major components allowed for a determination of the long-wavelength optical phonon characteristics, refractive indices dispersion, and parameters of interband and excitonic optical transitions.

Optical Hall effect-model description: tutorial.

The optical Hall effect is a physical phenomenon that describes the occurrence of magnetic-field-induced dielectric displacement at optical wavelengths, transverse and longitudinal to the incident electric field, and analogous to the static electrical Hall effect. The electrical Hall effect and certain cases of the optical Hall effect observations can be explained by extensions of the classic Drude model for the transport of electrons in metals. The optical Hall effect is most useful for characterization of electrical properties in semiconductors. Among many advantages, while the optical Hall effect dispenses with the need of electrical contacts, electrical material properties such as effective mass and mobility parameters, including their anisotropy as well as carrier type and density, can be determined from the optical Hall effect. Measurement of the optical Hall effect can be performed within the concept of generalized ellipsometry at an oblique angle of incidence. In this paper, we review and discuss physical model equations, which can be used to calculate the optical Hall effect in single- and multiple-layered structures of semiconductor materials. We define the optical Hall effect dielectric function tensor, demonstrate diagonalization approaches, and show requirements for the optical Hall effect tensor from energy conservation. We discuss both continuum and quantum approaches, and we provide a brief description of the generalized ellipsometry concept, the Mueller matrix calculus, and a 4×4 matrix algebra to calculate data accessible by experiment. In a follow-up paper, we will discuss strategies and approaches for experimental data acquisition and analysis.

Anisotropy, phonon modes, and free charge carrier parameters in monoclinicβ-gallium oxide single crystals

We derive a dielectric function tensor model approach to render the optical response of monoclinic and triclinic symmetry materials with multiple uncoupled infrared and farinfrared active modes. We apply our model approach to monoclinic $\beta$-Ga$_2$O$_3$ single crystal samples. Surfaces cut under different angles from a bulk crystal, (010) and ($\bar{2}$01), are investigated by generalized spectroscopic ellipsometry within infrared and farinfrared spectral regions. We determine the frequency dependence of 4 independent $\beta$-Ga$_2$O$_3$ Cartesian dielectric function tensor elements by matching large sets of experimental data using a point by point data inversion approach. From matching our monoclinic model to the obtained 4 dielectric function tensor components, we determine all infared and farinfrared active transverse optic phonon modes with $A_u$ and $B_u$ symmetry, and their eigenvectors within the monoclinic lattice. We find excellent agreement between our model results and results of density functional theory calculations. We derive and discuss the frequencies of longitudinal optical phonons in $\beta$-Ga$_2$O$_3$. We derive and report density and anisotropic mobility parameters of the free charge carriers within the tin doped crystals. We discuss the occurrence of longitudinal phonon plasmon coupled modes in $\beta$-Ga$_2$O$_3$ and provide their frequencies and eigenvectors. We also discuss and present monoclinic dielectric constants for static electric fields and frequencies above the reststrahlen range, and we provide a generalization of the Lyddane-Sachs-Teller relation for monoclinic lattices with infrared and farinfrared active modes. We find that the generalized Lyddane-Sachs-Teller relation is fulfilled excellently for $\beta$-Ga$_2$O$_3$.

Interband optical transitions of Zn

AbstractExperimental results of an optical study of single‐crystal zinc are presented. Components of the Zn dielectric function tensor were measured by spectroscopic ellipsometry in the 0.1–5 eV spectral range. In the NIR–VIS range, the dielectric function spectra show two clearly resolved, polarization‐dependent optical features located at about 1 and 1.7 eV. The optical features were analyzed in a framework of parallel‐band optical transitions. The performed theoretical calculations of the optical conductivity spectra well reproduce the experimental data with respect to positions, intensities, and polarization dependencies of the observed interband absorption peaks.

Anisotropic magneto-optical hysteresis of permalloy slanted columnar thin films determined by vector magneto-optical generalized ellipsometry

We present magneto-optical dielectric function tensor data of permalloy slanted columnar thin films obtained by vector magneto-optical generalized ellipsometry. Room-temperature hysteresis magnetization measurements were performed in transverse, longitudinal, and polar configurations, and at three different sample in plane azimuth orientations. We observe strongly anisotropic hysteresis behaviors, which depend on the nanocolumn and magnetizing field orientations. We present a model dielectric function approach by implementing vectorial magneto-optical magnetization hysteresis functions with major axes coincident with the major optical polarizability axes of the nanocolumns. Our approach provides excellent match with experimental data and reveals strongly anisotropic coercive field and ferromagnetic model parameters for the permalloy slanted columnar thin films. Ferromagnetic coupling is strongest parallel to the nanocolumn direction and substantially weaker perpendicular to the nanocolumns.

Dielectric function spectra and inter-band optical transitions in TlGaS2

TlGaS2 with a quasi-two-dimensional structure has been accessed by spectroscopic ellipsometry over the 1.5–6.0eV spectral range. A uniaxial approach applicable to monoclinic TlGaS2 at room temperature has been employed for ellipsometric data treatment. Principal components of the dielectric function tensor have then been retrieved. Inter-band optical transitions associated with the obtained dielectric function have been determined by using standard critical point analysis. The transitions have been assigned within the electronic band structure obtained for TlGaS2 from calculations based on density functional theory.

Electronic transitions and dielectric function tensor of a YMnO3 single crystal in the NIR-VUV spectral range

The dielectric function tensor elements of an YMnO3 single crystal yield M0 critical point like band-band absorption with discrete spectrally localized Lorentzian transitions within the band gaps.

Anisotropic Bruggeman effective medium approaches for slanted columnar thin films

Two different formalisms for the homogenization of composite materials containing ellipsoidal inclusions based on Bruggeman's original formula for spherical inclusions can be found in the literature. Both approximations determine the effective macroscopic permittivity of such an idealized composite assuming randomly distributed dielectric particles of equal shape and differ only in the definition of the depolarization factors. The two approaches are applied to analyze ellipsometric Mueller matrix spectra acquired in the visible and near-infrared spectral region from metal and semiconductor slanted columnar thin films. Furthermore, the effective dielectric function tensor generated by the two Bruggeman formalisms is compared to effective major axes dielectric functions individually determined with a homogeneous biaxial layer approach. Best-match model parameters of all three model approaches are discussed and compared to estimates from scanning electron microscope images. The structural parameters obtained from all three optical modeling approaches agree well with the electron microscopy technique. A comparative discussion is given for the validity and applicability of the three model approaches for analysis of future devices structures that may require optical readout using generalized ellipsometry methods.

Infrared dielectric anisotropy and phonon modes of rutile TiO2

Spectroscopic ellipsometry in the mid-infrared and far-infrared spectral range and generalized ellipsometry in the mid-infrared spectral range are used to investigate the anisotropic dielectric response of rutile TiO2. The ordinary and extraordinary dielectric function tensor components and all infrared active phonon mode parameters of single crystalline rutile TiO2 are determined with high accuracy for wavelengths from 3 μm to 83 μm. The data were acquired from samples of (001), (100), and (111) surfaces cut from bulk single crystals. A factorized model dielectric function is employed in order to determine the frequencies and damping parameters of the transverse and longitudinal phonon modes with A2u and Eu symmetries. The bands of total reflection of s- and p-polarized light in dependence of the angle of incidence for highly symmetric sample cuts and orientations are derived. Excellent agreement with phonon modes reported in literature is obtained. Introduction of two additional modes for ordinary as well as extraordinary component of the dielectric function tensor was necessary to most accurately match the experimental data. The spectral position of the additional modes is compared to the calculated phonon density of states. The low-frequency dielectric constants are calculated from the determined phonon mode parameters and the high-frequency dielectric constants by applying the Lyddanne-Sachs-Teller relation. The presented data revise existing infrared optical function data and will be suitable for interpretation of any kind of infrared spectra for bulk TiO2 single crystal substrates, thin films, and TiO2 nanostructures.