Impact of magnetism upon chemical interactions in Fe alloys

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1 Impact of magnetism upon chemical interactions in Fe alloys A.V. Ruban KTH, Stockholm 2009 M. Ohr, 1985 (Pettifor) C. Wolverton, 2005

Fundamental models Quantum mechanics Quantum theory Atomic level simulation Statistical mechanics Classical dynamics Force field Continuum

2 Multi-scale engineering design FEM CAD-CAM Engineering design Density functional theory: ca 100 atoms 1 nm, 0 K Kinetic Monte Carlo >1 million atoms, 20 nm, long times Molecular dynamics ca 10 nm 10-6 s Thermo-Calc DICTRA Phase Field Life time assessment Many years Fundamental models Quantum mechanics Quantum theory Atomic level simulation Statistical mechanics Classical dynamics Force field Continuum models Phenomenological models Macro simulation Micro structure Memika Size 1 (Å) 10 (Å) 100 (Å) 1 (µm) 1 (mm) Slide by P. Krozhavyi and J. Ågren

3 De Boer et al. Cohesion in Metals Sc Ti V Cr Mn Co Ni Y Zr Nb Mo Tc Ru Rh Pd Lu Hf Ta W Re Os Ir Pt U Pu Cu Ag Au

4 Outline 1. Origin of bonding and magnetism in TM 2. Magnetism and magnetic exchange interactions in Fe. 3. Effective cluster interactions (ECI) of chemical configurational Hamiltonian. 5. Separation and coupling of magnetic and chemical degrees of freedom: 5.1. Temperature dependent ECI: a) fcc Fe-Ni alloys; b) bcc Fe-Cr alloys 5.2. Local chemical environment effects in Fe-Cr alloys. 6. Conclusions and outlook.

5 Bonding and magnetism in TM In transition metals bonding and magnetism have a common origin: valence electron d-state Friedel model for bonding E Cohesive energy Vacancy formation W is the d-band width W D(E) d-band Surface energy

6 Stoner model of itinerant (band) magnetism Stoner criterion:

7 Ferromagnetic state in Fe Sws = 2.67 a.u. bcc fcc hcp m = 2.2 µb m = 1.18 µb m = 0 I = Ry (from Janak)

8 Impact of magnetism on physical properties

9 Atomic volume and bulk modulus of transition metals

10 Magnetic energies in bcc Fe 10 mry = ev = 1579 K Disordered local moment state is a state with randomly oriented local spins on atoms. It represents a hightemperature paramagnetic state of a Heisenberg-type magnet. In the case of non-relativistic systems, it is exactly mapped onto a random alloy of spin-up and spin-down states: A A

11 Paramagnetic (disordered local moment) state in Fe bcc fcc hcp m = 2.02 µb m = 1.67 µb m = 1.40 µb

12 Antiferromagnetic state (q=(001)) bcc fcc hcp m = 1.69 µb m = 1.51 µb m = 1.62 µb

13 Magnetic interactions and Hamiltonian 1. Heisenberg magnetic Hamiltonian: H H = J ijσ iσ j ij It assumes independence of interactions on the magnetic state and configuration. This is not the case of Fe and its alloys! E ss ( q) / sin 2 (θ) = J( q) However, it works reasonably well in many cases if cautiously used.

14 Curie temperature in bcc Fe FM DLM 1. FM interactions yield very poor results for Curie temperature. Renormalization helps, but not enough. DLM interactions work well (what was wrong with Oguchi's calculations???). 2. The electronic temperature and the choice of the xc-potential affects substantially the results for Curie temperature (error up to 10%). 3. There is non-negligible contribution to the Curie temperature from high order interactions, which in the LDA case lower T_c by about 50 K (the strongest four-site exchange parameter is for the tetrahedron of the 4 nn and 2 nnn ( type) -0.73meV).

15 Finite temperature magnetism In the LSDA only one-electron excitations are taken into consideration which give rise to the simplest Stoner type magnetic excitations between different spin bands at finite temperatures. At the same time, other type of magnetic excitations, leading for instance to longitudinal spin fluctuations, are important in Fe and its alloys at finite temperatures. A.I. Lichtenchtein, et al. PRL One of the solutions is to use a higher-level ab initio theory, which includes the corresponding many-body high-temperature excitations. Dynamical mean field theory (DMFT) seems to be one of the possible choices. However, DMFT is too cumbersome for materials science related problems and. it is still mean-field. Fortunately, this type of excitations play usually little role in the chemical bonding, although important for structural properties!

16 Atomic configurational thermodynamics in alloys

Cluster expansion of physical properties (Snachez, Ducastelle, Gratias 1984) binary A c B 1-c alloy σ i = +1 1 if i site occupied by A if i site occupied by B

17 Cluster expansion of physical properties (Snachez, Ducastelle, Gratias 1984) binary A c B 1-c alloy σ i = +1 1 if i site occupied by A if i site occupied by B Characteristic functions: Form a complete and orthonormal set, with the inner product: Any function of configuration, F(σ), can be expended in this set: where

18 Ising Hamiltonian: Grand Canonical The total energy of an alloy can be expended upon the basis of spin-variable products depends on alloy composition configurationally-dependent part For a finite system (N-atom) the effective cluster interactions are This is a strict definition for a system with finite range interactions.

19 Ising Hamiltonian: Canonical Ensemble In the case of infinite system at a fixed concentration, the product of spin-variable fluctuations δσi form complete and orthogonal basis for the cluster expansion. H = 1 2 p ij p V p (2),eff δσ i δσ j t ijk t V t (3),eff δσ i δσ j δσ k + Relationship between GCE and CE effective interactions GCE = CE interactions for c = 0.5 (σ=0) However, the latter is true only for the corresponding property if interactions are finite.

20 Do the interactions have some physical meaning (are the measurable)? In general NO, if we consider them as parameters of a model Hamiltonian. In this case their values will depend on how much of the phase space and in which way such a model Hamiltonian covers. (An example: concentration dependent versus concentration independent interactions). In particular YES, since effective interactions (or magnetic exchange interaction parameters), are related to non-local susceptibilities of the system determined for specific external and internal degrees of freedom): Where χ 0 is the susceptibility of non-interacting atoms (chemical interactions) or static on-site susceptibility (magnetic).

21 Diffuse scattering experiments The scattering intensity, I(q), is given by several contributions: Krivoglaz-Clapp-Moss formula: Nicholson et al. (J. Phys.: Cond. Mat., 2006) has recently proven that there is a unique correspondence between n-site interactions and n-site interactions for the order n Ising Hamiltonian. Within this formalism, interactions are really unique for a given alloy concentration and temperature.

22 Temperature (concentration,...) dependent interactions All type of interactions and (thermal) excitations affected by the alloy atomic configuration should contribute to the effective interactions: - electronic spectrum - magnetic state - atomic vibrations - strain-induced and other type of elastic interactions - coupled excitations... Direct calculations of effective interactions which include all these effects is practically impossible task. Note: If FP MD simulations are possible, it is not the case for FP MC simulations of alloy configuration!

23 Coarse-graining of partition function Here summation is over all possible (relevant) degrees of freedom: atomic configurational, electronic, vibrational and magnetic. However, they have different time scales: - configurational (the frequency of atomic jumps): 10-6 s infinity; - vibrational (the inverse of the Debye frequency): s; - magnetic (the inverse of spin-wave frequency): s; - electronic (for d-metals is given by inverse d-band with): s. Here: p is vibrational, m magnetic, and e electronic degrees of freedom.

Effective pair interactions in FM and PM states: Heisenberg system with one magnetic alloy component In FM state all magnetic atoms are indistinguishable, so if A is magnetic: V ij = V ij A B In

24 Effective pair interactions in FM and PM states: Heisenberg system with one magnetic alloy component In FM state all magnetic atoms are indistinguishable, so if A is magnetic: V ij = V ij A B In disordered local moment (DLM) model for paramagnetic binary alloy (A 0.5A 0.5) c B 1-c : On a very short time scale 3 different type of interactions (3-component system): V ij AB = 1 ( 2 V A B A ij + V A ) ;B A ij = V B ij 1 4 V A A ij = V ij A B + 2J ij xc

25 Chemical and magnetic interactions in FM and DLM fcc Cu 0.83 Mn 0.17 This is an almost ideal Heisenberg system!

26 What is the atomic configuration of the Fe-Ni Invar (Fe65Ni35)? V. Crisan, P. Entel, H. Ebert, H. Akai, D. D. Johnson, and J.B. Staunton, Phys. Rev. B 66, (2002) : Magnetochemical origin for Invar anomalies in iron-nickel alloys Chemical short- or long-range order and negative interatomic exchange interaction of electrons in antibonding majority-spin states force the facecentered-cubic lattice to compete simultaneously for a smaller volume (from antiferromagnetic tendencies) and a larger volume (from Stoner ferromagnetic tendencies). From some industrial heat treatment prescription: Annealing: Hold at 1450 F(790 C) 30 minutes for every inch of thickness, followed by air cooling. To obtain maximum dimensional stability, soak at 1500 F(815 C) and water quench. Reheat to 600 F(315 C) for 1 hour and air cool.

27 Invar cannot be found on the standard Fe-Ni phase diagram

28 More detailed structure and magnetic phase diagrams K.B. Reuter et al, 1987 from meteorites From M. Acet et al 1997

29 However, one can also find phase diagrams like this, where the existence of an ordered L1 2 -Fe 3 Ni can be found. a paper from 2005

30 Ferromagnetic enthalpies of formation in Fe-Ni (FP-DFT, Yu. Mishin et al 2005)

31 Impact of magnetic state on effective interactions in fcc Fe-Ni alloys 5 TABLE I. Ordering energies in mry/at. of Fe 50 Ni 50 alloy in the FM and DLM states obtained from the SGPM interactions and in the direct total energy calculations. (mry) FM DLM Structure DLM state FM state SGPM E tot SGPM E tot L V i (2) 1 CH Fe65Ni Coordination shell number FIG. 1. Color online Effective pair interactions in Fe 65 Ni 35 obtained for the ferromagnetic FM and paramagnetic DLM states. phase FM DLM L DO

Atomic short range order in Fe0.64Ni0.36 at 750 K J.L. Robertson et al. PRL 82, 1999 TABLE I. Warren-Cowley short-range order coefficients a lmn for Fe 63.2 Ni 36.8 Invar.

0062 Theory: lmn αlmn lmn αlmn 0 1.000 211 0.013 FIG. 2(color). Two dimensional plot of the short-range order scattering with contours in Laue units.

32 Atomic short range order in Fe0.64Ni0.36 at 750 K J.L. Robertson et al. PRL 82, 1999 TABLE I. Warren-Cowley short-range order coefficients a lmn for Fe 63.2 Ni 36.8 Invar. The numbers in parentheses are the standard deviations based on our best estimates of the total error. lmn a lmn lmn a lmn (53) (3) (2) Theory: lmn αlmn lmn αlmn FIG. 2(color). Two dimensional plot of the short-range order scattering with contours in Laue units. Intensities are reconstructed from the recovered a lmn s. The dark blue contour indicates the minimum (0.13) and red the maximum (1.83) value. The shape of the (100) diffuse peak along h 1 00 associated with the chemical short-range order intensity is shown in the inset. The sharp central peak and broad base of the profile are associated with the thickness and diameter of the platelets, respectively Ordering transition at 150 K

33 Effective pair interactions in FM state with reduced magnetization Global magnetization, m, 0 < m < 1 can be modeled by using partial disordered local moment description: (A 1-y A y ) 1-c B c, where y is the relative fraction of Fe atoms having the spindown orientation. When m = 1 2y. V ij AB = 1 + m 2 V A B ij + 1 m 2 V A B ij + 2(1 m 2 AA )J ij This means that effective interactions should exhibit non-linear dependence on the magnetization

34 Effective interactions in Fe-Cr: theory and experiment (I. Meribeau, et al Phys. Rev. Lett. 53 (1984) 687) diffuse-neutron-scattering experiments for Fe-5, 10, 15 at.% Cr alloys: V 1 changes sign at about 15 at. % Cr.

35 Temperature (magnetization)-dependent interactions in Fe 90 Cr 10 Ruban et al. Phys. Rev. B (2008)

36 Order-disorder transition in Fe3Ni (L12) Tc exp = 790 K Calculations by Marcus Enholm (LIU)

37 Magnetic frustrations in Fe-Cr alloys Interactions at 1st c.s.: Fe-Fe ferromagnetic Fe-Cr antiferromagnetic Cr-Cr antiferromagnetic This means that there is a frustration, when additional Cr atoms appear in the alloy. It is resolved by decreasing magnetic moment of Cr, and thereby decreasing Fe-Cr exchange interactions.

38 Magnetic moment of Cr in random Fe Cr An almost real random alloy: 432-atom supercell, FM state. The local magnetic moment of Cr atoms in alloy strongly depends on the number of n.n. Cr atoms. This means that some Cr are very different from the others.

Local environment effect on the chemical interactions in random alloy in 432-atom supercell (SGPM-LSGF calculations) There is no more simple meaning of Fe-Cr chemical interaction: The way Fe atoms

39 Local environment effect on the chemical interactions in random alloy in 432-atom supercell (SGPM-LSGF calculations) There is no more simple meaning of Fe-Cr chemical interaction: The way Fe atoms interact with Cr atoms depend on the chemical local environment of both Fe and Cr atoms. The chemical bond is determined by d-states of each individual atom, but they are modified by the magnetic state, which is sensitive, in its turn, to the local environment.

40 Conclusions In this talk only magnetic thermal excitations in the simples form have been considered. However, it is enough to make a firm conclusion, that there exist quite serious problems in a simple microscopic consideration of alloys, given in this particular case by an Ising-like model, where chemical identity is only restricted by the chemical symbol. This means, for instance, that in the multiscale consideration of the problem, there exist very serious problem not only in the interconnection between atomistic and continuos models, but in connection between electronic structure and atomistic level as well. This also means that we a way too far from realistic atomistic description of many Fe alloys at some particular external conditions.

41 41

EFFECTIVE MAGNETIC HAMILTONIANS: ab initio determination

ICSM212, Istanbul, May 3, 212, Theoretical Magnetism I, 17:2 p. 1 EFFECTIVE MAGNETIC HAMILTONIANS: ab initio determination Václav Drchal Institute of Physics ASCR, Praha, Czech Republic in collaboration