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

Phenomenon of magnetostriction.. “Para-process” Re-orientation Applied field reduces thermal disorder orienting moments along field. It is the source of the volume magnetostriction in all ferromagnets 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

Thermal expansion anomaly of Invar-type. ω lat – thermal expansion due to lattice vibrations, usually well follows Gruneisen law. ω m – magnetic contribution, which vanishes in paramagnetic state. It exists in all metallic magnets ω s0 – spontaneous volume magnetostriction for Fe-based Invar ω s0 ~1-2% ω s0 (Fe-Ni) ~ 2.1% Magnetic contribution to the thermal expansion can be viewed as sort of spontaneous paraprocess.

“classical” Invar alloys Fe-NiRECo 2 Experimental examples.

After M. Shiga and Y. Nakamura J.Phys. Soc. Jap. (1979) (Zr 1-x Nb x )Fe 2 – Laves Phase compounds.

Modern measurements at low T: example (Er-Y)Co 2 ω s0 – need not to be large if T c is small. R. Hauser et al. Phys. Rev. B 61, 1198 (2000) Modern measurements at low T reveals a lot of materials with Invar type anomaly Which has a small T c.

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

s-d model Fixed Spin Moment calculations for YCo2 F d is the Helmholtz potential of the itinerant subsystem “Simple” example 1: RECo 2 Laves Phases.

Stoner Co bandSpin-fluctuations included S. Khmelevskyi, P.Mohn, JMMM, (2003)525

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

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 T c. 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, T c (and Gruneisen coefficient).

Fe-based transition metal alloys. Classical Invar systems: fcc Fe-Ni, Fe-Pt, Fe-Pd (technical Invars). bcc Fe-Co AFe 2 – 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.

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: A 1-x B x  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 T c

Disordered Fe-Pt Fe 74 Pt 26 :  so (exp) =1.7% ;  so (calc) =1.9% S. Khmelevskyi, I. Turek, P. Mohn, PRL, (2003) TB-LMTO, LDA spd-basis Partial DLM calculations T c =ΓM loc 2 /(3k B )

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)

e/a=8.6 e/a=9 Fe 70 Pt 30 Fe 50 Pt 50 Transition from strong ferromagnetic state at T=0K to the weak ferromagnetic state in PM region Why local Fe moment anomalously decreases in Invar alloy composition?

Ordered and disordered Fe 3 Pt Cu 3 Au - structure Long-Range Order parameter: where c(Fe) is total concentration of Fe in the alloy and c I (Fe) is a concentration of Fe atoms on Pt sites.

Ordered and disordered Fe 3 Pt Khmelevskyi, Ruban, Kakehashi, Mohn, Johansson, PRB 72 (2005) Bulk KKR-ASA, spdf basis, and Muffin tin electrostatic corrections Spontaneous magnetostriction moderately decreases with increasing of ordering

Fe 65 Ni 35 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 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 Crisan et al. Phys. Rev. B 66, (2002)

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 Fe 65 Ni 65 become antiferromagnetic at volumes lower then experimental ones Antiferromagnetic scenario cannot be ruled out

Wang et al. JAP (1998) What is magnetic ground state of Fe 65 Ni 35 ? Fe 65 Ni 65 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)

Problem number 2: LSDA

Fe 65 Ni 35 INVAR alloy. GGA, Full-Charge Density EMTO results:  so (exp)=2.2% ;  so (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)

antiferromagnetic scenario: contra-example. (Zr 1-x Nb x )Fe 2 – Laves Phase compounds. and YFe 2 – non-Invar system.

R ws (FM) a.u. R ws (DLM) a.u. ω s0 (cal./exp.) % m Fe (FM), μ B m Fe (DLM), μ B m A (FM), μ B ZrFe / Zr 0.7 Nb 0.3 Fe / YFe / ~ KKR-ASA, spdf basis with MT-corrections, GGA Calculations with DLM.

Zr 0.7 Nb 0.3 Fe 2

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.