Cij Values Research Articles

A comprehensive DFT investigation of inorganic halide perovskites GaXCl3 (X = Ca, Sr, and Ba) for optoelectronics application

In recent times, ABX3 halide perovskite materials have emerged as revolutionary components in photovoltaic solar cells, functioning as photoabsorbers. Recently, there has been significant research attention directed towards lead halide perovskites. Nevertheless, there is ongoing research to identify lead-free alternatives due to the toxicity of lead. Therefore, the present study utilizes density functional theory to investigate the physical characteristics of the lead-free halide perovskites GaXCl3 (where X represents Ca, Sr, and Ba). The study aims to assess the feasibility of GaXCl3 (X = Ca, Sr, and Ba) for use in optoelectronic applications through an analysis of its structural, electrical, optical, and mechanical properties. The band gaps of GaCaCl3, GaSrCl3, and GaBaCl3 compounds have been found to be 3.45 eV, 3.22 eV, and 2.92 eV, respectively, which correspond to higher indirect band gaps. Additionally, elastic constants were computed in order to gain insight into the mechanical characteristics of the substance. The results reveal that all three compounds exhibit positive elastic constant Cij values, demonstrating their mechanical stability. However, their phonon dispersion curves show some negative parts which indicate their dynamic instability with temperature. Considerable performance in optoelectronic technology can be deduced from the absorption and optical conductivity of ultraviolet and visible light.

Computational investigation of the structural, elastic, electronic, and thermodynamic properties of chloroperovskites GaXCl3 (X = Be, Ca, or Sr) using DFT framework

In this study, we employed the pseudopotential plane wave approach to examine the influence of the X atom (X = Be, Ca, or Sr) on the physical properties of isostructural chloroperovskites GaXCl3. The GGA-PBEsol functional was employed to simulate the exchange–correlation interactions. The computed equilibrium lattice parameters exhibit a high level of concordance with the existing theoretical findings. The cohesion energy and enthalpy of formation were computed to verify the energetic stability of the materials under consideration. The determined values of the single-crystal elastic constants (Cij) indicate that GaBeCl3 remains mechanically stable up to a hydrostatic pressure of 18 GPa. Similarly, GaCaCl3 preserves its stability up to 5 GPa, while GaSrCl3 remains mechanically stable up to 1.25 GPa. The projected Cij values were used to estimate several elastic moduli and related properties, including the shear and bulk moduli, sound wave speeds, Young’s modulus, Poisson’s ratio, and Debye temperature. The energy band structures of the studied compounds, as predicted by the HSE06 functional, demonstrate their wide bandgap semiconductor nature. Specifically, GaBeCl3 demonstrates an indirect bandgap of 3.828 eV, while GaCaCl3 reveals an indirect bandgap of 4.612 eV and GaSrCl3 has an indirect bandgap of 4.405 eV. The quasiharmonic Debye approach was employed to examine various thermal parameters, including the temperature dependence of the unit cell volume, bulk modulus, expansion coefficient, Debye temperature, isochoric and isobar heat capacities, Grüneisen parameter, and entropy function. It has been shown that GaBeCl3 demonstrates a lower thermal expansion coefficient and a higher Debye temperature in comparison to GaCaCl3 and GaSrCl3.

Ab initio predictions of pressure-dependent structural, elastic, and thermodynamic properties of CaLiX3 (X = Cl, Br, and I) halide perovskites

In this comprehensive investigation, we undertook an ab initio exploration of the pressure-dependent structural, elastic and thermodynamic attributes of lithium-based halide perovskite compounds, namely CaLiCl3, CaLiBr3 and CaLiI3. Our analytical approach encompassed a diverse set of parameters, and the main conclusions and implications of our study are summarized as follows: (i) Calculated values of formation enthalpy and cohesion energy were determined for these perovskite compounds. Our results notably affirm the structural and thermodynamic stability of these materials in their cubic lattice configuration. (ii) Our optimized network parameters, derived from ab initio calculations, demonstrated commendable congruence with previously established theoretical predictions. This agreement strengthens the credibility of our conclusions. (iii) Using strain-constraint methodology, we successfully estimated the single-crystal elastic constants (Cij) of these compounds. This data served as the basis for further analysis. (iv) Using the obtained Cij values, we calculated a complete suite of elastic moduli for CaLiX3 (X = Cl, Br and I) in polycrystalline aggregates. This encompassed bulk modulus, Young's modulus, shear modulus, Lame coefficients, Poisson's ratio and Debye temperature, thus providing valuable information on the mechanical behavior of materials. (v) Using Debye's quasi-harmonic approach, we systematically investigated the temperature dependencies of several essential thermodynamic properties. These include the lattice parameter, thermal expansion coefficient, bulk modulus, Debye temperature, and isochoric and isobaric heat capacities. These analyzes covered a wide temperature range while maintaining fixed selected pressures. Overall, our study aims to provide the scientific community with a robust and comprehensive dataset regarding the structural, elastic, and thermodynamic attributes of CaLiX3 perovskite compounds.

The use of unmixing technique in stream sediment geochemical exploration

This research has aimed at employing unmixing method as a sediment tracing technique and its use in stream sediment geochemical exploration. Sediment tracing could be applied to determine the relative contributions of source units (Cij) to the metal contents of stream sediment samples. This is the first attempt to use the “fingerprinting” technique for estimating the background values of stream sediment samples.In catchment basin analysis of stream sediment data, the relative area sizes of lithological units (Xij) were used as the proxies for their contributions to the metal contents of stream sediment samples. The Hughes mixing model was solved using the particle swarm optimization (PSO) method to assign the contributions of 7 lithological units to 69 sample catchment basins. The results revealed some similarities between the estimated Cij values and the relative area sizes of lithological units (Xij) in all catchment basins, demonstrating the representativeness of area sizes as the proxies for the relative contributions of upstream sources to the metal content of stream sediments. The background values at the basin outlet were estimated using both Xij and Cij values. The results, therefore, revealed the superiority of the mixing model in estimating the background values at the basin outlet for most of the elements. The mixing model accurately estimated the elemental concentrations of Fe, Mo, Sb, Th, Ti, Y, and Yb; however, the unmixing approach failed to accurately predict the metal contents of elements such as Cr, Ni, Pb, Li, and Zn.

The single-crystal elastic properties of the jadeite-diopside solid solution and their implications for the composition-dependent seismic properties of eclogite

Abstract The 13 single-crystal adiabatic elastic moduli (Cij) of a C2/c jadeite sample close to the ideal composition (NaAlSi2O6) and a natural P2/n diopside-rich omphacite sample have been measured at ambient condition by Brillouin spectroscopy. The obtained Cij values for the jadeite sample are: C11 = 265.4(9) GPa, C22 = 247(1) GPa, C33 = 274(1) GPa, C44 = 85.8(7) GPa, C55 = 69.3(5) GPa, C66 = 93.0(7) GPa, C12 = 84(1) GPa, C13 = 66(1) GPa, C23 = 87(2) GPa, C15 = 5.4(7) GPa, C25 = 17(1) GPa, C35 = 28.7(6) GPa, C46 = 14.6(6) GPa. Voigt-Reuss-Hill averaging of the Cij values yields aggregate bulk modulus KS = 138(3) GPa and shear modulus G = 84(2) GPa for jadeite. Systematic analysis combing previous single-crystal elasticity measurements within the diopside-jadeite solid solution indicates that the linear trends are valid for most Cij values. The νp and νs of omphacite decrease with diopside content, though the velocity changes are small as diopside component exceeds 70%. We also found that both the isotropic νp and νs, as well as the seismic anisotropy of eclogite, changed strongly with the bulk-chemical composition. The relationship between the anisotropic velocities of eclogite and the chemical composition can be a useful tool to trace the origin of the eclogitic materials in the Earth's mantle.

Elastic constants of single-crystal Pt measured up to 20 GPa based on inelastic X-ray scattering: Implication for the establishment of an equation of state

Phonon velocities and densities for Pt were measured based on inelastic X-ray scattering from ambient pressure to 20GPa in order to independently determine its equation of state (EOS). Phonon velocities were determined with sine dispersion relations. Cij values were obtained by fitting phonon velocities and densities to the Christoffel equation. We found that the obtained Cijs were in good agreement with previously reported Cijs at ambient condition. Based on the Cij values in various conditions, experimental pressures were calculated. The EOS of Pt as a primary pressure scale was determined based on the experimental pressures. We report K’=5.17 with fixed KT=274.1GPa and V0=60.360 Å3 for Vinet EOS. Our scale is in good agreement with several previously published scales based on shock experiments and XRD.

First-principle calculations of the elastic properties of [formula omitted] semiconductors

First-principle calculations of the elastic properties of AIIBIVC2V semiconductors have been performed using the plane wave pseudo-potential method within the local density approximation. The values of elastic stiffness constants Cij, i.e., C11, C33, C44, C66, C12, C13; bulk modulus (B) and shear modulus (G) have been calculated. The Debye temperature (ΘD), Young’s modulus (Y), Poisson’s ratio (υ), Zener anisotropic factor (A) and G/B ratio have also been estimated. The values of Cij of ZnSnAs2 have been calculated for the first time. The calculated values of these parameters are compared with the available experimental and theoretical values. Reasonably good agreement has been obtained between them.

Effect of carbon concentration on shear modulus of (W1/2Al1/2)CZ

Using density functional theory (DFT), we have studied the effect of carbon concentration Z on the shear modulus G, elastic constant cij, and Poisson ratio ν of carbides (W1/2Al1/2)CZ while the most stable structures are determined by calculating the lowest formation energy. It is found that the maximal G value and the lowest Poisson ratio ν value are reached at Z = 3/4 due to atomic configuration changes, additionally, G values of (W1/2Al1/2)CZ should be lower than that of WC due to lower cij values. The results of electronic analysis show that the increased covalent bonding of C-2p and W-5d contributes to anisotropic cij and G positively and results in a lower ν value because the excess negative charge induced by the substitution of Al3+ for W4+ is balanced by decreasing C atoms.

The elastic properties of diopside, CaMgSi2O6

The 13 single-crystal elastic moduli of two samples of diopsidic pyroxenes close to the ideal composition (CaMgSi2O6) from different locations have been measured at ambient pressure and room temperature by Brillouin spectroscopy. We obtain (in GPa, ±1σ uncertainty): C11 = 229.0(4), C22 = 179.0(4), C33 = 242.5(4), C44 = 78.9(3), C55 = 68.1(2), C66 = 78.2(3), C12 = 78.0(7), C13 = 69.8(6), C23 = 58.0(7), C15 = 9.9(3), C25 = 6.1(5), C35 = 40.9(3), C46 = 6.6(2), and C11 = 226.1(9), C22 = 179.5(8), C33 = 239.2(9), C44 = 78.1(6), C55 = 69.2(4), C66 = 76.4(8), C12 = 77.4(10), C13 = 70.2(13), C23 = 56.7(16), C15 = 9.9(7), C25 = 5.9(12), C35 = 41.0(7), C46 = 6.8(4) for the two diopside samples. There are no major differences in the elastic tensor of the nearly pure diopside composition compared with the second sample containing slightly higher Al and Fe contents. Polycrystalline averaging of the Cij for the aggregate bulk and shear moduli yield Ks = 114.6(7) GPa, G = 72.7(4) GPa and Ks = 113.7(8) GPa, G = 72.2(5) GPa for the two diopside samples. The shear moduli reported here are 8% larger than those determined from previous measurements on a similar natural sample. We confirm the existence of systematic correlations between the composition of the M2 site and most of the Cij values. No unusual compositional dependence of Cij values or the shear modulus near end-member diopside composition is observed.

Advanced resonant ultrasound spectroscopy for measuring anisotropic elastic constants of thin films

ABSTRACTThis paper presents an advanced resonant ultrasound spectroscopy (RUS) method to determine the elastic constants Cij of thin films. Polycrystalline thin films often exhibit elastic anisotropy between the film growth direction and the in‐plane direction, and they macroscopically show five independent elastic constants. Because all of the Cij of a deposited thin film affect the mechanical resonance frequencies of the film/substrate layer specimen, measuring resonance frequencies enables one to determine the Cij of the film with known density, dimensions and the Cij of the substrate. Resonance frequencies have to be measured accurately because of low sensitivity of the Cij of films to them. We achieved this by a piezoelectric tripod. Mode identification has to be made unambiguously. We made this measuring displacement–amplitude distributions on the resonated specimen surface by laser Doppler interferometry. We applied our technique to copper thin film and diamond thin film. They show elastic anisotropy and the Cij smaller than bulk values of Cij. Micromechanics calculations indicate the presence of incohesive bonded regions.

Acoustic microscopy measurement of elastic constants by using an optimization method on measured and calculated SAW velocities: effect of initial cij values on the calculation convergence and influence of the LFI transducer parameters on the determination of the SAW velocity

Abstract The determination of elastic constants in anisotropic solids by line-focus acoustic microscopy (LFAM) is discussed. The velocity of leaky surface acoustic waves (SAWs) has been obtained for different directions of propagation ( φ ) from the V ( z ) curves. The elastic constants are determined from the comparison of measured and calculated velocities. The principle is based on the minimization of the summation of the square difference ∑| V Φ c − V Φ m | 2 , by using a Simplex method. This paper deals with the case of cubic-crystallite solids defined by three independent elastic constants. The influence of the initial values of c ij on the final result is discussed in the case of MgO substrate and TiN deposited on MgO substrate. Experimental SAW velocity profile is measured on (100) silicon substrate as a function of φ between [100] and [010] directions. The influence of the transducer parameters and defocusing distance are also discussed.

Anomalous Elastic Properties of Si/Ge Superlattices: The Role of Interfaces

The elastic constant tensors of Sim/Gen superlattices with different modulation wavelengths have been determined by Brillouin light scattering spectroscopy, and their Young's moduli measured by the nanoindentation technique. Two sets of samples have been studied: m = n and m = 4n. The C33 elastic constant obtained for Sin/Gen superlattices (where bilayers are built-up by an equal number of Si and Ge monolayers) are some 13% above the predictions of the continuum elasticity theory, using the Cij values of bulk Si and Ge. On the other hand, the elastic constants tensor of Si4n/Gen superlattices (where bilayers are built-up by four times more Si monolayers than Ge ones) matches perfectly the predictions. A thorough structural analysis by low-angle X-ray reflectivity, high-angle diffraction and X-ray absorption spectroscopy has been carried out and a deep characterization of the microstructure is presented here. The unexpected value of C11 in Sin/Gen superlattices can be attributed to the compressed Si–Ge alloy found at the wide interfaces present in that set of superlattices.

Influence of the short-range structural properties on the elastic constants of Si/Ge superlattices

The elastic constant tensor of Sim / Gen superlattices with different modulation wavelengths has been determined by Brillouin light scattering spectroscopy. The C11 elastic constant obtained for Sin / Gen (where bilayers are built up by an equal number of Si and Ge monolayers) and Sin / Ge4n superlattices are 13% above the predictions of the continuum elasticity theory, using the Cij values of bulk Si and Ge. On the other hand, the elastic constant tensor of Si4n / Gen and Si3n / Gen superlattices matches perfectly the predictions. A structural analysis by extended x-ray absorption fine structure spectroscopy of the interfaces has been carried out and the unexpected different elastic behaviour can be explained by the observed strains at the superlattice interfaces.

Single-crystal elastic constants of MoSi2 with the C11b structure

Single-crystal elastic constants of MoSi2 have been measured by the rectangular parallelepiped resonance (RPR) method over the temperature range 4–1373 K. The values at room temperature agree with those reported by Nakamura et al. [1], and the values extrapolated to 0 K are in good agreement with theoretical calculates by Alouani et al. [2]. Values of cij except for c12 and c13 decrease with increasing temperature while those of c12 and c13 are virtually temperature independent. Elastic property of the compound is discussed in terms of the strength of atomic bonds by adopting the approach introduced by Musgrave and Pople [3]. The elastic constants are expressed with three parameters corresponding to first-nearest-neighbor central, first-nearest-neighbor non-central and second-nearest-neighbor non-central interactions. The extent of contribution of the directional nature in atomic bonding to the elastic properties of MoSi2 is discussed in the light of the temperature variation of the three parameters we introduced.

Elastic properties of the incommensurate phase of akermanite, Ca2MgSi2O7

The crystal structure of akermanite, Ca2Mg-Si2O7, consists of mixed tetrahedral sheets formed by [MgO4] tetrahedra and [Si2O7] groups interleaved along the c axis with Ca2+ ions in eight-fold coordination. Above 358 K, the structure is tetragonal \((P\bar 42_1 {\text{ }}m)\), and below it is incommensurate with modulations parallel to [110] and \([1\bar 10]\). The elastic stiffness moduli, Cij of the incommensurate phase at room temperature were measured from wave velocities in the 20–75 MHz carrier frequency range by the ultrasonic phase comparison method using optically clear synthetic single crystal plates (3×3×2 mm) oriented parallel to (100), (001), (110) and (101) planes. The Cij values (GPa) are: C11 159.40, C33 149.43, C44 30.26, C66 58.10, C12 76.58 and C13 57.80. In (010) and (001) planes, the compressional modulus, ρV2(L) from the longitudinal wave, L is considerably larger than the shear moduli, ρV2(T1, T2) both from the in-plane and perpendicular-to-plane shear waves, T1 and T2. The relatively small values of the shear moduli indicate the ease of tetrahedral rotations in response to in-plane and perpendicular-to-plane shears and may provide preconditions for structural changes involving shear-type atomic movements.

A heuristic proǵramminǵ procedure for sequencinǵ the static flowshop

Abstract This paper describes a heuristic procedure for sequencing the n job m machine static flowshop. Basically, the procedure is performed in two overall steps. In the first step, each of the n jobs is tested as a potential immediate follower to each of the other jobs. In effect, this step of the procedure asks the question, ‘how well does a particular job fit in terms of job blocking or machine idleness if it were to follow some other job?’ An overall figure of merit, or cost cij, is determined for each job j as a follower to another job i. Six different heuristics are presented for determining sets of cij values. Using these values of cij, the second step then heuristically develops a job sequence by solving the travelling salesman problem. The paper also presents computational experience with the algorithm for a variety of randomly generated test problems (up to 50 jobs and 50 machines in size), and compares its performance with other published heuristic techniques.

Characteristic Debye temperature of ammonium fluoroberyllate, (NH4)2BeF4

From the elastic stiffness constants Cij of ammonium fluoroberyllate at room temperature, wave velocity surfaces have been calculated and are shown in three different sections (XY, YZ, ZX). The mean sound velocity in the crystal has thus been determined and used to calculate the Debye temperature of (NH4)2BeF4. The Debye temperature so determined is 381 K. Some of the bulk elastic properties of polycrystalline (NH4)2BeF4 have also been calculated using the above Cij values.

Electromagnetic acoustic resonance in metals and alloys

A technique has been developed for measuring the temperature dependence of elastic moduli of metals and alloys using electromagnetic excitation of acoustic resonance in small spheres polished optically smooth. A frequency swept marginal oscillator is used to generate and detect resonances without physical contact to the sample. Values of Cij are reproducible with accuracies of ∼1/104. Variations of ∼1/106 are detected using an oscillator mode which is phase locked to sample resonance. Measurements of C44 and (1/2)(C11−C12), from 300 to 4.2 K, on a 4.7-mm-diam. tungsten sphere are given to illustrate the technique.

The Substituent Effect. XI. Solvolysis of 5-, 6-, and 7-Substituted 1-(1-Naphthylethyl) Chlorides

Abstract Fourteen 5-, 6-, and 7-substituted 1-(1-naphthylethyl) chlorides were prepared and the solvolysis rates were determined in 80% (v/v) aqueous acetone at 45 °C. The effects of −R substituents at respective positions were treated on the basis of the equation, logk⁄k0=ρiσi+ρx+σx+=ρ(Cijσi+qrij+σπ+). The position dependency of the inductive effect was given by Cij(=ρi⁄ρi,4α); C3α=1.37, C4α=1.00, C5α=0.75, C6α=0.57, and C7α=0.72. The Cij values appear to be correlated with Dewar’s simplified field function 1/rij. The position dependency of Pi-electronic effect given by the ratio ρπ+⁄ρi,4α=qr·ij+ was consistent with the prediction from the MO indices, such as Forsyth’s Δqij(ArCH2+ parameters. The same treatment was applied also to three other naphthalene reactivities, detritiation, pKa of α-naphthoic acids and pKa of naphthylammonium ions. The linear ρi–ρi relation holds among these reactions, suggesting a reaction-independent scheme of the inductive transmission. The qr·ij+ values vary at conjugate positions with reaction but not at non-conjugate positions irrespective of reactions. The results are discussed in comparison with those of relevant treatments.

Temperature and pressure dependence of the elastic properties of RbAg4I5

An ultrasonic pulse superposition technique was used to measure the velocities of the three pure-mode waves which propagate along [110] in a single crystal of rubidium tetra-silver pentaiodide. The temperature was varied from 25 °C, through a phase transformation at −65 °C, and to a second phase transformation at −151 °C. There was a peak in the ultrasonic attenuation at the −65 °C transition for the longitudinal and one of the shear waves, and the attenuation and velocity of all of the waves were completely reversible upon cycling the temperature through this transition. The lower-temperature phase was generally several percent softer elastically than the room-temperature phase. Attempts to make measurements through the −151 °C transition were not successful. The pressure dependence of the wave velocities were determined at 25 °C. The adiabatic second-order elastic constants and their temperature and pressure derivatives at 25 °C which were determined from these measurements are as follows. The values of Cij (1011 dyn/cm2), C−1ij (∂Cij/∂T) (10−4 °K−1), and ∂Cij/∂P for C11 are 1.648±0.002, −3.54±0.12, and +8.73±0.11, respectively; for C12, 0.934±0.002, −3.73±0.13, and +6.29±0.11, respectively; and for C44, 0.4892±0.0005, −4.17±0.05, and +0.884±0.015, respectively. The Debye temperature of 90 °K and the thermal expansion coefficient of 0.566×10−4 °K−1 calculated from these values agree very well with values determined by other means.