Theory of magnetostriction in Invar materials.

1. Theory of magnetostriction in Invar materials. S. Khmelevskyi Center for Computational Materials Science, Vienna University of Technology P. Mohn, A. V. Ruban, I. Turek

2. Phenomenon of magnetostriction. . Applied field reduces thermal disorder orienting moments along field. It is the sourceof the volume magnetostriction in all ferromagnets Re-orientation “Para-process” Applied field induces a band splitting. Small effect. It may be ignored for ferromagnets. H=100T ~ 1mRy However it is the source of para-process at T = 0K Even in non-magnetic materials! Re-orientation of the magnetization along the field spin-orbit coupling anisotropy Anomalous thermal expansion in Invar

3. Thermal expansion anomaly of Invar-type. ωlat – thermal expansion due to lattice vibrations, usually well follows Gruneisen law. ωm – magnetic contribution, which vanishes in paramagnetic state. It exists in all metallic magnets ωs0 – spontaneous volume magnetostriction for Fe-based Invar ωs0 ~1-2% ωs0(Fe-Ni) ~ 2.1% Magnetic contribution to the thermal expansion can be viewed as sort of spontaneous paraprocess.

4. Experimental examples. “classical” Invar alloys Fe-Ni RECo2

5. (Zr1-xNbx)Fe2 – Laves Phase compounds. After M. Shiga and Y. Nakamura J.Phys. Soc. Jap. (1979)

6. Modern measurements at low T: example (Er-Y)Co2 ωs0 – need not to be large if Tc is small. Modern measurements at low T reveals a lot of materials with Invar type anomaly Which has a small Tc. R. Hauser et al. Phys. Rev. B 61, 1198 (2000)

7. Some Thermodynamics. Vonsovskyi, Shur (1948) Bean, Rodbell (1962) local moment model -effective exchange constant

8. “Simple” example 1: RECo2 Laves Phases. Fixed Spin Moment calculations for YCo2 s-d model Fdis the Helmholtz potential of the itinerant subsystem

9. V / V D -3 1 ´ 10 = w DyCo 2 HoCo 2 0 20 40 60 80 100 120 140 T(K) Stoner Co band Spin-fluctuations included S. Khmelevskyi, P.Mohn, JMMM, 272-276 (2003)525

10. “Simple” example 2: hcp Gadolinium. TB-LMTO Disordered Local Moment. Partial DLM Gd(4f-up)1-xGd(4f-down)x f-electrons – open core (m = 7mB). Thermal expansion A. Lindbaum and M. Rotter S.Khmelevskyi, I. Turek, P. Mohn, PRB 70, (2003)

11. Intermediate conclusions. • Invar anomaly is magnetovolume effect related to the spontaneous volume magnetostriction due to change of magnetization with temperature (very trivial). • Such effect exists in ALL magnetic materials (even in ALL non-magnetic in applied external field). • One need just to explain why in given Invar material such a contribution large enough to compensate (or be comparable in size to !) a thermal expansion due to lattice vibration in the temperature interval from 0K to Tc. • There is no intrinsic and unique feature of Invar materials, which is absent in “normal” ones. The difference between them only in quantity - related size of spontaneous magnetostrictions, Tc (and Gruneisen coefficient).

12. Fe-based transition metal alloys. • Classical Invar systems: fcc Fe-Ni, Fe-Pt, Fe-Pd • (technical Invars). • bcc Fe-Co • AFe2 – Laves Phases. • Problems. • One cannot separate a system into electronic subsystems, one of which is responsible for the intrinsic temperature dependent molecular field acting on another subsystem with anomalous magnetostrictive properties. • We should have working approach to tackle with finite temperature magnetism of itinerant magnets in intermediate regime between local and weak itinerant cases (our choice is DLM – “Do it better if …”). • Lots of additional material dependent complications: chemical disorder, antiferromagnetic interactions on frustrated lattice etc. • 50 years of intensive theoretical development on Fe-Ni, which are very complex alloy.

14. Disordered Local Moments state. Model for paramagnetic state: disordered alloy of the same sort of atoms with spin-up and spin down randomly distrubuted over the lattice sites. (Cyrot, Gyorrfy) Alloy analogy: A1-xBx  A()1-x A()x binary metallic alloy partially ordered state with local moments Using Coherent Potential Approximation A()0.5 A()0.5 – model paramagnetic state above Tc

15. Disordered Fe-Pt TB-LMTO, LDA spd-basis Partial DLM calculations Tc=ΓMloc2/(3kB) Tc=ΓMloc2/(3kB) Tc=ΓMloc2/(3kB) Fe74Pt26: wso(exp)=1.7% ; wso(calc)=1.9% S. Khmelevskyi, I. Turek, P. Mohn, PRL, 91 037201 (2003)

16. fcc Fe-Pd, fcc Fe-Pt and bcc Fe-Co Maximum of the spontaneous magnetostriction corresponds to the maximum of the local Fe moment drop in paramagnetic state. Khmelevskyi and Mohn, PRB 69 (2004)

17. Why local Fe moment anomalously decreases in Invar alloy composition? Fe70Pt30 Fe50Pt50 e/a=9 e/a=8.6 Transition from strong ferromagnetic state at T=0K to the weak ferromagnetic state in PM region

18. Ordered and disordered Fe3Pt Cu3Au - structure Long-Range Order parameter: where c(Fe)is totalconcentration of Fe in thealloy andcI(Fe)is aconcentration of Fe atomson Pt sites.

19. Ordered and disordered Fe3Pt Bulk KKR-ASA, spdf basis, and Muffin tin electrostatic corrections Spontaneous magnetostriction moderately decreases with increasing of ordering Khmelevskyi, Ruban, Kakehashi, Mohn, Johansson, PRB 72 (2005)

20. Nothing is new since there is exist nothing new. Fe-Ni case. Akai and Dederichs, PRB B 47, 8739 (1993) consistent with TB-LMTO calculations with spd-basis Fe65Ni35 KKR-ASA LSDA spd-basis D. D. Johnson, F. J. Pinski, J. B. Staunton, B. L. Gyorffy, and G.M. Stocks, in Physical Metallurgy of Controlled ExpansionInvar-type Alloys, 1990 Crisan et al. Phys. Rev. B 66, 014416 (2002)

21. Problem number 1: exchange interactions in Fe-Ni Inter-atomic exchange interactions of Heisenberg Hamiltonian calculated using Lichtenstein Green function formalism (Magnetic Force Theorem). GGA results. INVAR alloy Fe65Ni65 become antiferromagnetic at volumes lower then experimental ones Antiferromagnetic scenario cannot be ruled out

22. What is magnetic ground state of Fe65Ni35? Fe65Ni65 alloy Calculations with Local self-consistent Green Function methods (LSGF) 512 atoms super cell Moment of Fe atoms with 11 and 12 Fe nearest neighbors oriented anti-parallel to the total magnetization. Ruban, Khmelevskyi, Mohn, Johansson PRB 76 (2007) Wang et al. JAP (1998)

23. Problem number 2: LSDA

24. Fe65Ni35 INVAR alloy. GGA, Full-Charge Density EMTO results: wso(exp)=2.2% ; wso(calc)=3.2% Calculations of effective inter-atomic chemical interactions and MC simulations shows that Fe-Ni cannot be considered as partially ordered alloy. Short-Range order effects is also very weak. Ruban, Khmelevskyi, Mohn, Johansson PRB 76 (2007)

25. antiferromagnetic scenario: contra-example. (Zr1-xNbx)Fe2 – Laves Phase compounds. and YFe2 – non-Invar system.

26. Calculations with DLM. KKR-ASA, spdf basis with MT-corrections, GGA

27. Zr0.7Nb0.3Fe2

28. Zr0.7Nb0.3Fe2

29. Conclusions. • The Invar effect has common origin in all Invar-type magnetic systems. Spontaneous volume magnetostriction, which large enough to compensate thermal expansion. It should be only RELATIVELY large. • In all considering cases the source of large magnetostriction is decrease of the size of the local moments induced by the thermal disorder of magnetic moments. • The source of this decrease may be different in different Invars. • The difference between Invar and non-Invar systems is quantitative – not qualitative.