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Computational method
Before constructing the crystal structures of the off-stoichiometric Ni50MnxIn50-x alloys, the preferred site of the excess Mn atoms with an amount of 6.25 at. % in a unit cell with 16 atoms was determined by the formation energies using the Vienna Ab-initio Simulation Package (VASP) [25]. The conventional cell of the cubic L21 Ni50Mn25In25 alloy is presented in Fig. 3.1. In the figure, the
primitive cell is also outlined with the dashed lines. Here, the atomic combination of the primitive cell (4 atoms) is defined as a motif, and it is used as one of the structure units for calculating the formation energy and total magnetic moment of other structures (tetragonal and orthorhombic, as described later). The full charge density method was applied to treat the tetragonal distortion by the Extra Muffin-Tin Orbital method combined with the Coherent Potential Approximation (EMTO-CPA) [26–28] with the optimization of the magnetic structure. For the phase stability analysis of the tetragonal L10 martensite of the Ni-Mn-In alloys with different Mn concentrations, the more efficient EMTO-CPA was used. For the simulation of the orthorhombic martensite structures with different Mn concentrations, a cell containing 96 atoms was constructed with the supercell method allowing volume relaxation using the VASP. The Generalized Gradient Approximation (GGA), established by Perdew, Burke, and Ernzerhof (GGA-PBE) [29], was employed to describe the exchange-correlation functions. The electronic configurations were Ni (3d84s2), Mn (3d64s1), and In (4s24p1), respectively. Based on the convergence test, the reciprocal-grid density of 12 × 12 × 12 Gamma centered Monkhorst-Pack k-point meshes for the L21 primitive cell and 420 eV as the kinetic energy cutoff were applied in the VASP calculations. The EMTO was performed with a basis set including the s, p, d, f orbitals. After carefully testing the convergence of the equilibrium parameter against the k-point meshes, a 13 × 13 × 13 k-point meshes was set for the primitive cell with 4 atoms. The formation energy (Ef) of the Ni2MnxIn2-x was determined according to the following function of the Mn concentration (x): E f ( x ) E0 ( Ni2 Mn x In2x ) 2* E Ni x * E Mn (2 x ) * EIn.
Computational method: theory and methodology
In the framework of the Density Functional Theory (DFT) [19], the plane-wave basis projector augmented wave (PAW) method with the generalized-gradient approximation (GGA-PBE) [20] was used in the exact muffin-tin orbital method combined with the coherent potential approximation (EMTO-CPA) [21]. The ferromagnetic interactions of atoms were set for both austenite and martensite phases. Although the martensite phase has two kinds of structures (the modulated martensite with monoclinic structure, and the non-modulated martensite with tetragonal structure), only the non-modulated L10 structure was used to calculate the martensite phase stability, as the modulated structure was considered to consist of twin related L10 structures (the so-called adaptive structure) [22]. The convergence criterion of energy was 10-6 eV. A 13 × 13 × 13 k-point sampling mesh was set for the ferromagnetic austenite and martensite with 4 atoms, respectively. The lattice parameters of ground state for the two phases with different Mn contents were determined by the equation of states (EOS) with the Morse function [12] and were used as the equilibrium configurations for the further finite temperature calculations. According to the experimental observations [1,23], the extra Mn atom prefers to substitute In and stays at In site.
To determine the thermodynamic properties of crystalline materials, the temperature and volume related formulation of the Helmholtz free energy [15,24,25] should be used, as follows: F(V,T) E (V)F (V,T) F (V,T) F (V,T)T *S conf (T ) (4.1).
In the equation, E0K is the static lattice total energy at 0 K and is only volume dependent. Fvib, Fel and Fmag, are the vibrational, electronic excitation and magnetic free energies, respectively, and they are both volume and temperature dependent. Sconf is the configurational entropy. The vibrational contribution was simulated in the frame of the efficient quasi-harmonic Debye-Grüneisen model using the Gibbs2 code [26–28]. The electronic excitation contribution was calculated with the EMTO [21] using the finite temperature Fermi distribution [29,30]. Here, we only calculated the magnetic moment at finite temperatures to predict the variation of the magnetic entropy at finite temperatures using the EMTO [21]. The temperature dependent magnetic moment was derived from the magnetic free energy ( Fmag (T , )TSmag () ), where the magnetic entropy was estimated by the mean-field approach ( Smag ( ) kB log(i1) , in whichi is the magnetic moment of atom i) [15,31–34].
Entropy change during the martensitic transformation can be defined as [14]: S tr . S mar . S aus. (4.2).
Since the martensitic transition is diffusionless in the Ni-Mn-In alloys, the change of the configurational entropy is zero [35], and the anharmonic contributions is also negligible. Therefore, only the three main contributions should be taken into consideration for the Ni-Mn-In alloys [36– 38]: SS S ele S mag (4.3).
Results and discussion
Based on the equilibrium energies of the austenite and the NM martensite, we first analyzed the effect of Mn concentration on the ground state stability in terms of the static lattice contribution at 0 K which is the most important part of the conventional contribution to the free energy. Fig. 4.1 presents the equilibrium energy difference (∆E = Emar. – Eaut.) between the two phases in the ferromagnetic state as a function of Mn content at 0 K. It is seen that a positive value (0.026 eV) is reached at 25 at. % Mn, which indicates that the austenite is more stable than the martensite in the stoichiometric alloy. Apart from this, the energy differences of all compositions with excess Mn are negative, i.e. the total energy of the martensite is lower than that of the austenite at 0 K. This suggests that the ground state of the off-stoichiometric Ni50MnxIn50-x alloys is the martensite. Moreover, with the increase of the Mn content, the energy differences between the two phases decrease monotonously. This indicates that the stability of the NM martensite becomes higher with the increasing Mn content. This tendency is in good accordance with what predicted from the phase diagram of Ni-Mn-In alloys [2,7] which showed that the NM martensite with L10 structure is stable at high Mn concentration. Such a tendency was also observed in the experiments, e.g. the Ni50Mn50 alloys have a tetragonal structure at room temperature [4] and the martensitic transition temperature of the ferromagnetic martensite increases dramatically with the extra Mn [5,39]. However, the total energies obtained in the present work were calculated with the atoms fixed at their equilibrium positions at 0 K. In reality, the martensitic transitions of the Ni-Mn-In alloys were observed at different temperatures below or above room temperature [5]. Therefore, the temperature related entropy changes due to thermal excitation are essential to understand the phase transitions of the off-stoichiometric Ni50MnxIn50-x alloys.
Effect of effective Coulomb and exchange parameters on phase stability, structural and magnetic properties of Ni50Mn25In25 alloys
The effect of the Hubbard U and J coupling parameters on the phase stability, structural and magnetic properties of the ferromagnetic austenite with a cubic L21 structure in the stoichiometric Ni50Mn25In25 alloy were simulated with the Vienna Ab-initio Simulation Package (VASP) in the frame of the density-functional theory using the plane-wave pseudopotential calculations. The the phase stability of the ordered Ni50Mn25In25 austenite with the effective Coulomb and exchange U and J parameters was firstly investigated in the present work. The effect of the effective Coulomb and exchange U and J parameters on the optimized structural parameters, the phase stability (including the ground state energy, the formation energy and the elastic constants), and the electronic structure (including the magnetic moment, the band structure and the density of states and the charge density difference) of the cubic stoichiometric Ni50Mn25In25 austenite were obtained. The results of the first principle calculations with the GGA + U approach of the cubic L21 austenite of Ni50Mn25In25 alloy were as below.
i) The optimized lattice parameters vary with the effective Coulomb and exchange U and J parameters around the experimentally evaluated value. With all the J values, the lattice parameters a become extremely close at U equal to 6. The ground state energy arrives at a stable stage when J equal to 0.5. The elastic constants and the bulk modulus change with U and J coupling, whereas they do not show any evident tendency to affect the phase stability. Based on the optimization of the lattice parameter and the magnetic moment and the minimum ground state energy variation for further physical properties analysis, the effective Coulomb and exchange parameters of U = 6 eV and J = 0.5 were chosen for further calculation.
ii) With the exchange-correlation GGA + U (U = 6 eV and J = 0.5) method, the lattice parameter of the austenite increases slightly. Compared with the calculation without considering the effective Coulomb and exchange parameters, the magnetic moment increase slightly. However, the calculation showed a higher formation energy, which means that the consideration of the Hubbard U and J coupling reduced the stability of the L21 structure.
iii) The effects of the effective Coulomb and exchange parameters on the electronic structure were obtained by band structure, density of states and charge density difference. With some slight influence on the spin coupling, no obvious change was detected around the
Fermi Level. With the effective Coulomb and exchange parameters U and J coupling, the bandgap still shows the metallic bonding in the cubic austenite Ni50Mn25In25 alloy.
We first tried to systematically calculate the physical properties of the cubic austenite of the Ni50Mn25In25 alloy by considering the effective Coulomb and exchange U and J parameter into the exchange correlation by the GGA using the VASP with an aim to modulate more accurately for the metal transition element Ni and Mn with 3d orbitals. Since there is no obvious effect on the equilibrium crystal structure or the electronic structure, we could deduce that there is no need to add the efficient Coulomb parameter into the physical property simulation of the Heusler typed Ni-Mn-In alloys by the first-principle calculations.
Table of contents :
Chapter 1 Introduction
1.1 Ni-Mn-In ferromagnetic shape memory alloys
1.1.1 Crystal structure of Ni-Mn-In alloys
1.1.2 Magnetic properties of Ni-Mn-In alloys
1.2 Introduction of a quantum theory of materials
1.2.1 Schrödinger Equation
1.2.2 Variational principle of ground state
1.2.3 Hartree-Fock approximation
1.3 Fundamentals of the Density Functional Theory
1.3.1 Electron density
1.3.2 the Thomas-Fermi model
1.3.3 the Hohenberg-Kohn theorems
1.3.4 the Kohn-Sham equations
1.3.5 Methods for solving the Kohn-Sham Equation
1.3.6 Exchange-correlation functions
1.4 Motivation and objectives
Chapter 2 Effect of effective Coulomb and exchange parameters on phase stability, structural and magnetic properties of Ni50Mn25In25 alloys
2.1 Outline
2.2 Introduction
2.3 Computational method
2.4 Results and discussion
2.5 Conclusions
Reference
Chapter 3 Composition dependent martensitic structural preference of Ni50MnxIn50-x alloys by VASP and EMTO-CPA method
3.1 Outline
3.2 Introduction
3.3 Computational method
3.4 Results and discussion
3.5 Conclusions
References
Chapter 4 Composition dependent thermal excited contributions to phase stability of ferromagnetic Ni50MnxIn50-x alloys at finite temperatures
4.1 Outline
4.2 Introduction
4.3 Computational method: theory and methodology
4.4 Results and discussion
4.4 Conclusions
Reference
Chapter 5 Conclusions and perspectives
Publication list
Acknowledgement
