Lattice relaxations in disordered Fe-based materials in the paramagnetic state from first principles

In this work we propose a method for the structural relaxation of magnetic materials in the paramagnetic regime, in an adiabatic fast-magnetism approximation within the disordered local moment (DLM) picture in the framework of density functional theory (DFT). The method is straight forward to implement using any $ab$ $initio$ code that allows for structural relaxations. We illustrate the importance of considering the disordered magnetic state during lattice relaxations by calculating formation energies and geometries for an Fe vacancy and C insterstitial atom in bcc Fe as well as bcc Fe$_{1-x}$Cr$_x$ random alloys in the paramagnetic state. In the vacancy case, the nearest neighbors to the vacancy relax towards the vacancy of 0.16 {\AA} (-5% of the ideal bcc nearest neighbor distance), which is twice as large as the relaxation in the ferromagnetic case. The vacancy formation energy calculated in the DLM state on these positions is 1.60 eV, which corresponds to a reduction of about 0.1 eV compared to the formation energy calculated using DLM but on ferromagnetic-relaxed positions. The carbon interstitial formation energy is found to be 0.41 eV when the DLM relaxed positions are used, as compared to 0.59 eV when the FM-relaxed positions are employed. For bcc Fe$_{0.5}$Cr$_{0.5}$ alloys, the mixing enthalpy is reduced by 5 meV/atom, or about 10%, when the DLM state relaxation is considered, as compared to positions relaxed in the ferromagnetic state

Efficient and accurate determination of lattice-vacancy diffusion coefficients via non equilibrium ab initio molecular dynamics

We revisit the color-diffusion algorithm [P. C. Aeberhard et al., Phys. Rev. Lett. 108, 095901 (2012)] in nonequilibrium ab initio molecular dynamics (NE-AIMD), and propose a simple efficient approach for the estimation of monovacancy jump rates in crystalline solids at temperatures well below melting. Color-diffusion applied to monovacancy migration entails that one lattice atom (colored-atom) is accelerated toward the neighboring defect-site by an external constant force F. Considering bcc molybdenum between 1000 and 2800 K as a model system, NE-AIMD results show that the colored-atom jump rate k_{NE} increases exponentially with the force intensity F, up to F values far beyond the linear-fitting regime employed previously. Using a simple model, we derive an analytical expression which reproduces the observed k_{NE}(F) dependence on F. Equilibrium rates extrapolated by NE-AIMD results are in excellent agreement with those of unconstrained dynamics. The gain in computational efficiency achieved with our approach increases rapidly with decreasing temperatures, and reaches a factor of four orders of magnitude at the lowest temperature considered in the present study

A Test of Sovereignty: Franchise Tax Board of the State of California v. Gilbert P. Hyatt

In Franchise Tax Board of California v. Hyatt, the Supreme Court considers whether to overrule Nevada v. Hall, a 1979 Supreme Court decision. Hall permitted a State to be haled into the court of another State without its consent. In 2016, an evenly divided Supreme Court affirmed Hall 4-4 when faced with the same question, and following a remand to the Nevada Supreme Court, the Court has granted certiorari on this question once again. This Commentary contends that Hall was wrongly decided and should be overruled. The Constitution’s ratification did not alter the status of common-law State sovereign immunity, leaving intact not only State sovereign immunity in a State’s own court but also a State’s immunity to suits in the courts of another State without consent. However, this case, in which the Petitioner has already appeared in the court of another State, is not the appropriate vehicle for overruling Hall. State sovereign immunity should be restored at the next possible opportunity, when a State properly asks a federal court to enforce its common-law immunity from the courts of a sister State. Sovereigns should enjoy immunity not only in their own courts, but also in the courts of their peers

A theoretical study of disorder and decomposition in multinary nitrides hard coatings materials

The development of multinary nitrides materials has revolutionised the hard coatings industry over the last 20 years. Especially important materials systems in this matter have been TiAlN and CrAlN which shows higher hardness, better oxidation resistance and can perform at higher temperatures as compared to TiN. When synthesised through physical vapour deposition techniques these system form cubic rock salt structured Ti1-xAlxN and Cr1-xAlxN solid solutions as a metastable phase over a large part of the concentration range. One of the main objectives during the optimisation of the coatings has been to increase the amount of Al in the coating while still keeping the rock salt structure, avoiding phase separation and the formation of hexagonal wurtzite AlN. However, in Al-rich TiAlN coatings it was found that isostructural decomposition within the cubic phase was in fact beneficial for the coatings performance at working temperatures just below 1000 °C. The reason was that the formation of strained coherent c-AlN domains within the grains initiated a age-hardening of the coating. In the CrAlN coatings it is possible to solve a larger amount of Al in the cubic phase. Possibly connected to this fact is that no isostructural decomposition and in principle no age hardening has been observed in rock salt structured Cr1-xAlxN. Although large series of experimental investigations have been performed on these systems, no systematic theoretical study has yet been undertaken. This theoretical work is an attempt by means of first-principles calculations together with thermodynamics considerations within the framework of alloy theory, to close or decrease the knowledge gap between experimental observations of cutting performance of various coatings and the fundamental quantum mechanical and thermodynamics processes that governs it. We first consider the structural properties of the treated mixed nitride systems. The concept of atomic misfit or volume difference, which is well known in the community is studied and the physics that leads to a positive deviation from Vegard's rule is revealed. Then the TiAlN and CrAlN systems are studied in detail. A clear connection between the development of the electronic structure with composition and the mixing enthalpy of the alloys, and thus the tendency for decomposition, is found for TiAlN. The importance of magnetic effects on the thermodynamics of mixing in CrAlN is established leading to a qualitative lower tendency for decomposition. Since the nitrogen composition can deviate substantially from perfect stoichiometry in these systems, a study of the influence of nitrogen vacancies on the decomposition pattern of TiAlN in the cubic phase is performed. The results imply that a presence of nitrogen sub-stoichiometry in Al-rich TiAlN will enhance the tendency for isostructural decomposition. The achieved results including those for the systems ScAlN and HfAlN are compared and discussed especially by considering volume misfit and electronic bandstructure effects as driving forces for coherent and incoherent decomposition

Finite temperature elastic constants of paramagnetic materials within the disordered local moment picture from ab initio molecular dynamics calculations

We present a theoretical scheme to calculate the elastic constants of magnetic materials in the high-temperature paramagnetic state. Our approach is based on a combination of disordered local moments picture and ab initio molecular dynamics (DLM-MD). Moreover, we investigate a possibility to enhance the efficiency of the simulations of elastic properties using recently introduced method: symmetry imposed force constant temperature dependent effective potential (SIFC-TDEP). We have chosen cubic paramagnetic CrN as a model system. This is done due to its technological importance and its demonstrated strong coupling between magnetic and lattice degrees of freedom. We have studied the temperature dependent single-crystal and polycrystalline elastic constants of paramagentic CrN up to 1200 K. The obtained results at T= 300 K agree well with the experimental values of polycrystalline elastic constants as well as Poisson ratio at room temperature. We observe that the Young's modulus is strongly dependent on temperature, decreasing by ~14% from T=300 K to 1200 K. In addition we have studied the elastic anisotropy of CrN as a function of temperature and we observe that CrN becomes substantially more isotropic as the temperature increases. We demonstrate that the use of Birch law may lead to substantial errors for calculations of temperature induced changes of elastic moduli. The proposed methodology can be used for accurate predictions of mechanical properties of magnetic materials at temperatures above their magnetic order-disorder phase transition.Comment: 1 table, 3 figure

Phase stability of Fe from first-principles: atomistic spin dynamics coupled with ab initio molecular dynamics simulations and thermodynamic integration

The calculation of free energies from first principles in materials is a formidable task which enables the prediction of phase stability with high accuracy; these calculations are complicated in magnetic materials by the interplay of electronic, magnetic, and vibrational degrees of freedom. In this work, we show the feasibility and accuracy of the calculation of phase stability in magnetic systems with ab initio methods and thermodynamic integration by sampling the magnetic and vibrational phase space with coupled atomistic spin dynamics-ab initio molecular dynamics (ASD-AIMD) simulations [Stockem et al., PRL 121, 125902 (2018)], where energies and interatomic forces are calculated with density functional theory (DFT). We employ the method to calculate the phase stability of Fe at ambient pressure from 800 K up to 1800 K. The Gibbs free energy difference between fcc and bcc Fe at zero pressure as a function of temperature is calculated carrying out thermodynamic integration over temperature on the energies at the DFT level from ASD-AIMD, using a reference free energy difference calculated in the paramagnetic state at temperatures much higher than the magnetic transition temperatures with thermodynamic integration over stress-strain variables with disordered local moment (DLM)-AIMD simulations. We show the importance of the magnetic ordering temperature of bcc Fe on the $\alpha$ to $\gamma$ structural transition temperature, whereas the $\gamma$ to $\delta$ transition is well reproduced independently of the exchange interactions. The Gibbs free energy difference between the two structures is within 5 meV/atom from the CALPHAD estimate, and both transition temperatures are reproduced within 150 K. The present work paves the way to free energy calculations in magnetic materials from first principles with accuracy in the order of 1 meV/atom

Origin of the anomalous piezoelectric response in wurtzite Scx_xAl1−x_{1-x}N alloys

The origin of the anomalous, 400% increase of the piezoelectric coefficient in Sc$_x$Al$_{1-x}$N alloys is revealed. Quantum mechanical calculations show that the effect is intrinsic. It comes from a strong change in the response of the internal atomic coordinates to strain and pronounced softening of C$_{33}$ elastic constant. The underlying mechanism is the flattening of the energy landscape due to a competition between the parent wurtzite and the so far experimentally unknown hexagonal phases of the alloy. Our observation provides a route for the design of materials with high piezoelectric response.Comment: 10 pages, 4 figures, accepted for publication in Phys. Rev. Let

Anharmonicity changes the solid solubility of an alloy at high temperatures

We have developed a method to accurately and efficiently determine the vibrational free energy as a function of temperature and volume for substitutional alloys from first principles. Taking Ti$_{1-x}$Al$_x$N alloy as a model system, we calculate the isostructural phase diagram by finding the global minimum of the free energy, corresponding to the true equilibrium state of the system. We demonstrate that the anharmonic contribution and temperature dependence of the mixing enthalpy have a decisive impact on the calculated phase diagram of a Ti$_{1-x}$Al$_x$N alloy, lowering the maximum temperature for the miscibility gap from 6560 K to 2860 K. Our local chemical composition measurements on thermally aged Ti$_{0.5}$Al$_{0.5}$N alloys agree with the calculated phase diagram.Comment: 4 pages, 5 figures, supplementary materia

Longitudinal spin fluctuations in bcc and liquid Fe at high temperature and pressure calculated with a supercell approach

Investigation of magnetic materials at realistic conditions with first-principles methods is a challenging task due to the interplay of vibrational and magnetic degrees of freedom. The most difficult contribution to include in simulations is represented by the longitudinal magnetic degrees of freedom (LSF) due to their inherent many-body nature; nonetheless, schemes that enable to take into account this effect on a semiclassical level have been proposed and employed in the investigation of magnetic systems. However, assessment of the effect of vibrations on LSF is lacking in the literature. For this reason, in this work we develop a supercell approach within the framework of constrained density functional theory to calculate self-consistently the size of local-environment-dependent magnetic moments in the paramagnetic, high-temperature state in presence of lattice vibrations and for liquid Fe in different conditions. First, we consider the case of bcc Fe at the Curie temperature and ambient pressure. Then, we perform a similar analysis on bcc Fe at Earth's inner core conditions, and we find that LSF stabilize non-zero moments which affect atomic forces and electronic density of states of the system. Finally, we employ the present scheme on liquid Fe at the melting point at ambient pressure, and at Earth's outer core conditions ($p \approx 200$ GPa, $T \approx 6000$ K). In both cases, we obtain local magnetic moments of sizes comparable to the solid-state counterparts.Comment: 12 pages, 12 figure