Shear Elastic Constant C44 Research Articles

Structure, stress, and mechanical properties of Mo-Al-N thin films deposited by dc reactive magnetron cosputtering: Role of point defects

In this work, the structural and mechanical properties of ternary Mo-Al-N alloys are investigated by combining thin film growth experiments and density functional theory (DFT) calculations. Mo1−xAlxNy thin films (∼300 nm thick), with various Al fractions ranging from x = 0 to 0.5 and nitrogen-to-metal (Al + Mo) ratio ranging from y = 0.78 to 1.38, were deposited by direct-current reactive magnetron cosputtering technique from elemental Mo and Al targets under Ar + N2 plasma discharges. The Al content was varied by changing the respective Mo and Al target powers, at a fixed N2 (20 SCCM) and Ar (25 SCCM) flow rate, and using two different substrate temperatures Ts = 350 and 500 °C. The elemental composition, mass density, crystal structure, residual stress state, and intrinsic (growth) stress were examined by wavelength dispersive x-ray spectroscopy, x-ray reflectivity, x-ray diffraction, including pole figure and sin2ψ measurements, and real-time in situ wafer curvature. Nanoindentation tests were carried out to determine film hardness H and elastic modulus EIT, while the shear elastic constant C44 was measured selectively by surface Brillouin light spectroscopy. All deposited Mo1−xAlxNy films have a cubic rock-salt crystal structure and exhibit a fiber-texture with a [001] preferred orientation. The incorporation of Al is accompanied by a rise in nitrogen content from 44 to 58 at. %, resulting in a significant increase (2%) in the lattice parameter when x increases from 0 to 0.27. This trend is opposite to what DFT calculations predict for cubic defect-free stoichiometric Mo1−xAlxN compounds and is attributed to variation in point defect concentration (nitrogen and metal vacancies) when Al substitutes for Mo. Increasing Ts from 350 to 500 °C has a minimal effect on the structural properties and phase composition of the ternary alloys but concurs to an appreciable reduction of the compressive stress from −5 to −4 GPa. A continuous increase and decrease in transverse sound velocity and mass density, respectively, lead to a moderate stiffening of the shear elastic constant from 130 to 144 GPa with increasing Al fraction up to x = 0.50, and a complex and nonmonotonous variation of H and EIT is observed. The maximum hardness of ∼33 GPa is found for the Mo0.81Al0.19N1.13 film, with nitrogen content close to the stoichiometric composition. The experimental findings are explained based on structural and elastic constant values computed from DFT for defect-free and metal- or nitrogen-deficient rock-salt MoAlN compounds.

Influence of vacancy on the mechanical and thermodynamic properties of IrAl3 compound: A first-principles calculations

The influence of vacancy on mechanical and thermodynamic properties of IrAl3 is systematically investigated by using first-principles calculations. Three different vacancies such as Ir-va, Al-va1 and Al-va2 are considered in detail. The vacancy formation energy, elastic constants, elastic properties, Debye temperature and electronic structure of IrAl3 with vacancies are calculated. These vacancies are stability at the ground state, and IrAl3 prefers to form Ir-va vacancy in comparison with Al-va vacancy. The calculated elastic constant C11 and shear elastic constant C44 of Al-va2 vacancy are larger than that of the perfect IrAl3. Although Ir-va and Al-va1 vacancies weaken the resistance to volume and shear deformation, Al-va2 vacancy remarkably improves the shear deformation resistance and elastic stiffness. It is concluded that the high shear modulus and Young’s modulus are originated from the removed Ir atom to enhance the localized hybridization between Al atom and the Ir atom. Finally, it is found that the Debye temperature of Al-va2 vacancy is also larger than that of perfect IrAl3.

First-principles calculation of the structural and elastic properties of ternary metal nitrides TaxMo1-xN and TaxW1-xN

First-principles pseudo-potentials calculations of the mixing enthalpy, of the lattice constants a0 and of the single-crystal elastic constants cij for ternary metal nitrides TaxMe1-xN (Me=Mo or W) alloys considering the cubic B1-rocksalt structure is carried out. For disordered ternary alloys, we employ the virtual crystal approximation VCA in which the alloy pseudopotentials are constructed within a first-principles VCA scheme. The supercell method SC is also used for ordered structures in order to evaluate clustering effects. We find that the mixing enthalpy still remains negative for TaxMe1-xN alloys in the whole composition range which implies these cubic TaxMo1-xN and TaxW1-xN ordered solid solutions are stable. We investigate the effect of Mo and W alloying on the trend of the mechanical properties of TaN. The effective shear elastic constant c44, the Cauchy pressure (c12-c44), and the shear to bulk modulus G/B ratio are used to discuss, respectively, the mechanical stability of the ternary structure and the brittle/ductile behavior in reference to TaN, MeN alloys. We determine the onset transition from the unstable structure to the stable one B1-rocksalt from the elastic stability criteria when alloying MeN with Ta. In a second stage, in the frame of anisotropic elasticity, we estimate by one homogenization method the averaged constants <Cij> of the polycrystalline TaxMe1-xN alloys considering the special case of an isotropic medium with no crystallographic texture.

Elastic and anelastic relaxations associated with phase transitions inEuTiO3

Elastic and anelastic properties of single crystal samples of EuTiO3 have been measured between 10 and 300 K by resonant ultrasound spectroscopy at frequencies in the vicinity of 1 MHz. Softening of the shear elastic constants C44 and 1 (C11 − C12 )b y∼20–30% occurs with falling temperature in a narrow interval through the transition point, Tc = 284 K, for the cubic-tetragonal transition. This is accounted for by classical coupling of macroscopic spontaneous strains with the tilt order parameter in the same manner as occurs in SrTiO3. A peak in the acoustic loss occurs a few degrees below Tc and is interpreted in terms of initially mobile ferroelastic twin walls, which rapidly become pinned with further lowering of temperature. This contrasts with the properties of twin walls in SrTiO3, which remain mobile down to at least 15 K. No further anomalies were observed that might be indicative of strain coupling to any additional phase transitions above 10 K. A slight anomaly in the shear elastic constants, independent of frequency and without any associated acoustic loss, was found at ∼140 K. It marks a change from elastic stiffening to softening with falling temperature and perhaps provides evidence for coupling between strain and local fluctuations of dipoles related to the incipient ferroelectric transition. An increase in acoustic loss below ∼80 K is attributed to the development of dynamical magnetic clustering ahead of the known antiferromagnetic ordering transition at ∼5.5 K. Detection of these elastic anomalies serves to emphasize that coupling of strain with tilting, ferroelectric, and magnetic order parameters is likely to be a permeating influence in determining the structure, stability, properties, and behavior of EuTiO3.

Structure and elasticity of phlogopite under compression: Geophysical implications

We investigated the response of the crystal structure, lattice parameters, and unit-cell volume of hydrous layered silicate phlogopite at conditions relevant to subduction zone settings. We have used first principles simulation based on density functional theory to calculate the equation of state and full elastic constant tensor. Based on the generalized gradient approximation, the full single crystal elastic constant tensor with monoclinic symmetry shows significant anisotropy with the compressional elastic constants: c11=181GPa, c22=185GPa, c33=62GPa, the shear elastic constants c44=14GPa, c55=20GPa, c66=68Ga, and c46=−6GPa; the off diagonal elastic constants c12=48GPa, c13=12GPa, c23=12GPa, c15=−16GPa, c25=−5GPa and c35=−1GPa at zero pressure. The elastic anisotropy of phlogopite is larger than most of the layered hydrous phases relevant in the subduction zone conditions. The shear anisotropy, AVS for phlogopite is ∼77% at zero pressure condition and although it decreases upon compression it remains relatively high compared to other hydrous phases relevant in the subduction zone settings. We also note that the shear elastic constants for phlogopite are relatively low. Phlogopite also has a high isotropic bulk VP/VS ratio ∼2.0. However, the VP/VS ratio also exhibits significant anisotropy with values as low as 1.49. Thus, phlogopite bearing metasomatized mantle could readily explain unusual VP/VS ratio as observed from seismological studies from the mantle wedge regions of the subduction zone.

Ab initio study on stacking sequences, free energy, dynamical stability and potential energy surfaces of graphite structures

Ab initio simulations have been performed to study the structure, energetics and stability of several plausible stacking sequences in graphite. These calculations suggest that in addition to the standard structures, graphite can also exist in AA-simple hexagonal, AB-orthorhombic and ABC-hexagonal type stacking. The free energy difference between these structures is very small (∼1 meV/atom), and hence all the structures can coexist from purely energetic considerations. Calculated x-ray diffraction patterns are similar to those of the standard structures for 2θ ⩽ 70°. Shear elastic constant C44 is negative in AA-simple hexagonal, AB-orthorhombic and ABC-hexagonal structures, suggesting that these structures are mechanically unstable. Phonon dispersions show that the frequencies of some modes along the Γ–A direction in the Brillouin zone are imaginary in all of the new structures, implying that these structures are dynamically unstable. Incorporation of zero point vibrational energy via the quasi-harmonic approximation does not result in the restoration of dynamical stability. Potential energy surfaces for the unstable normal modes are seen to have the topography of a potential hill for all the new structures, confirming that all of the new structures are inherently unstable. The fact that the potential energy surface is not in the form of a double well implies that the structures are linearly as well as globally unstable.

Unusually large shear wave anisotropy for chlorite in subduction zone settings

AbstractUsing first principle simulations we calculated the elasticity of chlorite. At a density ρ~ 2.60 g cm−3, the elastic constant tensor reveals significant elastic anisotropy: VP ~27%, VS1 ~56%, and VS2 ~43%. The shear anisotropy is exceptionally large for chlorite and enhances upon compression. Upon compression, the shear elastic constant component C44 and C55 decreases, whereas C66 shear component stiffens. The softening in C44 and C55 is reflected in shear modulus, G, and the shear wave velocity, VS. Our results on elastic anisotropy at conditions relevant to the mantle wedge indicates that a 10–20 km layer of hydrated peridotite with serpentine and chlorite could account for the observed shear polarization anisotropy and associated large delay times of 1–2 s observed in some subduction zone settings. In addition, chlorite could also explain the low VP/VS ratios that have been observed in recent high‐resolution seismological studies.

Evolution of thermal properties from graphene to graphite

We report on the evolution of thermal properties from graphene to graphite as a function of layer thickness and temperature. The onset of the inter-layer compressional elastic constant C33 and the shear elastic constant C44 results in a large difference between the magnitudes and temperature dependencies of the specific heat and in-plane lattice thermal conductivity of bi-layer graphene (BLG) and single-layer graphene. The changes between BLG and few-layer graphene (FLG) decrease with increase in the number of layers. The cross-plane lattice thermal conductivity increases almost linearly with the number of layers in ultra-thin FLG.

Alpha to omega martensitic phase transformation pathways in pure Zr

Abstract This first principles study attempts to find the lowest enthalpy barrier transformation pathway (TP) for pressure induced α → ω transformation in pure Zr by systematically studying TPs suggested experimentally along with TPs generated from symmetry analysis. We find that symmetry generated pathways and model pathways given from experimental inputs essentially outline the same subset of possible pathways. The most favorable TP, comprising of smaller strains and atomic shuffles compared to others, has a C2/c as a common space group along the TP with enthalpy barrier of 22 meV/atom at 0 K. It is evident from the study that the softening of shear elastic constant C44 and Γ point transverse optical phonon frequency as a function of pressure in α-Zr are associated with change in effective potential landscape for the E2g phonon at Γ point of the Brillouin zone. The calculated electronic band structure as a function of pressure and E2g phonon mode shows a similar topological change in the band structure which drives this transformation.

Surface acoustic waves and elastic constants of InN epilayers determined by Brillouin scattering

AbstractThe surface acoustic wave velocity in InN has been experimentally determined by means of Brillouin scattering experiments on c ‐ and m ‐face epilayers. From simulations based on the Green's function formalism we determine the shear elastic constants c66 and c44 and propose a complete set of elastic constants for wurtzite InN. The analysis of the sagittal and azimuthal dependence of the surface acoustic wave velocity indicates a slightly different elastic behavior of the m ‐face sample that basically affects the c44 elastic constant. (© 2012 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Elastic constants in tetragonal hen egg-white lysozyme crystals containing large amount of water

Transverse sound velocity of cross-linked tetragonal hen egg-white (HEW) lysozyme crystals containing large amount of water in the crystal was measured using ultrasonic pulse-echo method. All elastic constants of cross-linked crystals were observed to be C11=C22=5.50 GPa, C12=4.33 GPa, C13=C23=3.94 GPa, C33=5.22 GPa, C44=C55=0.68 GPa, and C66=0.84 GPa, respectively. We found that the elastic constants of the cross-linked crystals are identical to those of the intrinsic ones without cross-linking. Moreover, we found that tetragonal HEW lysozyme crystals that enclose large amount of water show decreased elastic constants (softening). In particular, the shear elastic constants C44=C55 and C66 showed more softening effect comparing with other elastic components.

Magnetoelastic coupling in Fe100−xGex single crystals with 4&amp;lt;x&amp;lt;18

In this paper we examine the elastic (c′ and c44) and magnetostrictive (λ100 and λ111) behaviors of Fe100−xGex for 4&amp;lt;x&amp;lt;18, quantities used further to find the fundamental magnetoelastic coupling constants b1 and b2 at room temperature. The x dependence of b1 and b2 for Fe100−xGex is contrasted to those of Fe100−xGax and Fe100−xAlx. While the rhombohedral shear elastic constant c44 is almost insensitive to the type and amount of solute, the tetragonal shear constant c′ shows a pronounced and rapid softening with increasing x for all three alloys but with different decreasing slopes. Similarly, while the rhombohedral magnetostriction λ111 behavior is analogous for all three alloy systems, showing a sign change from negative to positive at the onset of chemical order, the tetragonal magnetostriction λ100 behavior differs. For the Ga and Al alloys, λ100 maintains positive values over the entire x range, both curves showing large peak values, whereas λ100 of Fe100−xGex exhibits a moderate positive peak followed by a negative dip, both of comparable magnitude. Finally the tetragonal coupling constant −b1 of Fe–Ge shows a marked, sharp decrease as chemical order occurs at x∼12 at. % Ge. The decline continues until the ordered D03 phase is fully established at x∼18 at. % Ge. The peak value of |b1| for Fe–Ge is approximately half of those for Fe–Ga and Fe–Al. This smaller value of |b1|, obtained for the higher electron concentration Ge alloy, is consistent with predictions based on band structure calculations. The rhombohedral coupling constant −b2 shows a consistent sign change at the occurrence of chemical ordering in both Fe–Ga and Fe–Ge.

Ab initio study of the elastic properties of single and polycrystalTiO2,ZrO2 and HfO2 in the cotunnite structure

In this work, we study theoretically the elastic properties of the orthorhombic(Pnma) high-pressure phase of IV-B group oxides: titania, zirconia and hafnia. By means of theself-consistent SIESTA code, pseudopotentials, density functional theory in the LDA and GGAapproximations, the total energies, hydrostatic pressures and stress tensor components arecalculated. From the stress–strain relationships, in the linear regime, the elastic constantsCij are determined. Derived elastic constants, such as bulk, Young’s and shearmodulus, Poisson coefficient and brittle/ductile behavior are estimated with thepolycrystalline approach, using Voigt–Reuss–Hill theories. We have found thatC11,C22 and C33 elastic constants of hafnia and zirconia show increased strengthwith respect to the experimental values of the normal phase,P 21/c. A similar situation applies to titania if these constants are comparedwith its normal phase, rutile. However, shear elastic constantsC44,C55 and C66 are similar to the values found in the normal phase. This fact increasesthe compound anisotropy as well as its ductile behavior. The dependenceof unit-cell volumes under hydrostatic pressures is also analyzed.P–V data, fitted to third-order Birch–Murnaghan equations of state, provide the bulk modulusB0 and its pressurederivatives B′0. In this case, LDA estimations show good agreement with respect to recent measured bulk moduliof ZrO2 and HfO2.Thermo-acoustic properties, e.g. the propagation speed of transverse, longitudinal elasticwaves together with associated Debye temperatures, are also estimated.

Order parameters in the Verwey phase transition

The Verwey phase transition in magnetite is analyzed on the basis of the Landau theory. The free energy functional is expanded in a series of components belonging to the primary and secondary order parameters. A low-temperature phase with the monoclinic P2/c symmetry is a result of condensation of two order parameters X3 and Δ5. The temperature dependence of the shear elastic constant C44 is derived and the mechanism of its softening is discussed.

Crystalline smectic-B films as fluctuating systems: static and dynamic x-ray scattering.

Static and dynamic x-ray scattering has been used to characterize thermal fluctuations in free-standing smectic films in the crystalline-B phase. The results are remarkably similar to those in smectic-A films, in spite of the three-dimensional positional order in the crystalline-B phase. The main difference is the disappearance of the characteristic Landau-Peierls instability of the smectic-A phase. The dynamic nature of the fluctuations is explicitly demonstrated by x-ray photon correlation spectroscopy. This fluctuation behavior of crystalline-B films can be attributed to the small value of the shear elastic constant C44.

Elastic properties of GaSe films epitaxially grown on the Si(111)1 × 1-H surface, studied by Brillouin scattering

Brillouin light scattering has been used to study the elastic properties of GaSe films of different thicknesses epitaxially grown on a hydrogen-terminated non-reconstructed Si(111) surface. Both Rayleigh and Sezawa guided acoustic modes have been revealed, and their velocity dispersion measured as a function of the ratio between the film thickness and the acoustic wavelength. This enabled us to determine four of the five GaSe film elastic constants through a non-linear best fit procedure. The fifth elastic constant c66 was obtained from detection of shear horizontal waves in the thicker film. The results obtained show that the film constants are similar to those of the bulk material, except for the shear elastic constant c44 which is appreciably lower. The values of the elastic constants thus obtained were used to calculate the Brillouin cross-section taking into account the rippling induced by surface phonons both at the free surface and at the interface. This permitted us to achieve a satisfactory reproduction of the experimental spectra.

Effective elastic constants in EnZeZnSe1 − xSx strained-layer superlattices

In a Brillouin scattering study of ZnSeZnSe1 − xSx strained-layer superlattices' film, we have observed large increases for the longitudinal elastic constants C11, and significant decreases for shear elastic constants C44, than the predicted value seems to be only relevant to the existence of imperfect interfaces. In order to explain these results, we use an interface layer (I layer) model. The structure (-AIBIAIBI-) of superlattices has been both observed in transmission-electron microscopy (TEM) observations and Raman spectra.

Enhancement of the shear elastic constant in periodic and quasiperiodic Ta/Al superlattices

The Brillouin light-scattering technique has been exploited in order to reveal surface acoustic phonons in both periodic and Fibonacci quasiperiodic Ta/Al superlattices. In periodic specimens, about 0.5 μm thick, it is found that, as the superlattice period is reduced from 10 to 4 nm, the phase velocity of the Rayleigh acoustic mode exhibits an anomalous increase, corresponding to a 20% enhancement of the effective shear elastic constant c44. In 2k-component and 3k-component Fibonacci quasiperiodic superlattices, with total thickness of about 1.5 μm, an enhancement of c44 similar to that of periodic specimens has been evidenced, which is connected to the average density of the interfaces, rather than to the quasiperiodicity. This enhancement can be attributed to the presence of interfaces extended over a number of atomic planes, due to the mutual interdiffusion between Ta and Al, as inferred by x-ray and electron microscopy experiments.

Molecular Dynamics Simulation of Electron Irradiation-Induced Amortization of the Ordered Compound NiZr

ABSTRACTWe present the results of a molecular dynamics simulation of the crystalline-to-amorphous transformation in the ordered intermetallic compound NiZr, using embedded-atom potentials. It is found that NiZr can be amorphized by random atom exchanges as well as by the introduction of Frenkel pairs. In both cases, the shear elastic constants C44 and C decreased drastically and eventually became equal. The average shear elastic constant fell to about 40% of its original value. These results are in general agreement with experimental observations.

Contamination Effects in Mo/Ni Superlattices

AbstractA softening of the shear elastic constant c44 has been observed previously in Mo/Ni superlattices as a function of decreasing bilayer thickness below approximately 100 Å.[1] We have prepared a series of Mo/Ni superlattice films by ion beam sputtering doped with varying concentrations of either aluminum or oxygen. The chemical and structural properties of these films were then determined using x-ray diffraction (XRD) and Auger electron spectroscopy (AES). The shear elastic properties were characterized by measuring the surface acoustic wave (SAW) velocity of the deposited films. We demonstrate structural and elastic property effects resulting from Al and O impurity incorporation in Mo/Ni multilayers.