FIRST-PRINCIPLES CALCULATIONS OF THE INSTABILITIES IN Fe-(Ni, Co, Pt) ALLOYS

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1 FIRST-PRINCIPLES CALCULATIONS OF THE INSTABILITIES IN Fe-(Ni, Co, Pt) ALLOYS P. ENTEL a,,h.ebert b, V. CRISAN b and H. AKAI c a Theoretical Low-Temperature Physics, Gerhard Mercator University, Duisburg, Germany b Institute of Physical Chemistry, Ludwig-Maximilians University, Mnchen, Germany c Department of Physics, Osaka University Osaka , Japan (Received...) Abstract First-principles calculations using the Korringa-Kohn-Rostocker method and the Coherent-Potential Approximation for Fe-Ni, Fe-Ni- Co and Fe-Pt alloys show that several features are responsible for the Invar anomalies. Atomic short range ordering in the alloys is responsible for the appearance of antiferromagnetic and non-colinear magnetic moments. The antiferromagnetic contributions are responsible for two effects, the negative anharmonicity due to the tendency of the alloys to have a smaller lattice constant, as well as the tendency to have a larger lattice constant because of additional density of states of antibonding majority-spin orbitals at the Fermi level, which simultaneously stabilizes the antiferromagnetic moments. Keywords: Invar, Korringa-Kohn-Rostocker Method, Coherent-Potential Approximation, Atomic short-range order Corresponding author: Tel ; fax: , entel@thp.uni-duisburg.de 1

2 1. INTRODUCTION It has recently been observed in supercell calculations for Fe 66 Ni 34 that noncolinear (NC) magnetism in the disordered alloy leads to further negative lattice anharmonicity, small bulk modulus, negative pressure derivative of the bulk modulus and a smearing of the magnetovolume instability (van Schilfgaarde, Abrikosov and Johansson, 1999), which has been argued to be at the origin of the Invar effect (Mohn, 1999). Very early Johnson et al. had discussed this kind of effects theoretically by using the concept of atomic short-range order (ASRO) and the disordered local-moment theory (Johnson, Pinski, Staunton, Gyrffy et al., 1990; Johnson and Shelton, 1997). Non-collinearity (NC) or in a much simpler picture, antiferromagnetic (AF) contributions in the Invar alloys has always been believed to be an important ingredient. On the basis of early neutron scattering experiments on Fe 63 Ni 37 (Dubinin, Sidorov, Teploukhov and Arkhipov, 1973) it was inferred that an AF contribution of the ( 1, 1, 0) type develops at liquid helium 2 2 temperatures which was already discussed in the frame of a NC magnetic structure. The importance of antiferromagnetic (AF) contributions was recognized in most experimental work and was sometimes used together with the two-states model of Weiss (Weiss, 1963) to interpret the experiments. For instance, anomalies of magnetostriction and thermal expansion coefficient of Fe 65.5 Ni 34.5 measured in an external magnetic field were interpreted in terms of AF fcc Fe clustering in the ferromagnetic (FM) matrix (Zähres, Acet, Stamm and Wassermann, 1988). Pronounced AF Invar and anti-invar behavior was observed in the Fe-Mn alloys by (Schneider, Acet, Wassermann and Pepperhoff, 1995). For an overview of further experimental work and references we refer to (Wassermann, 1990, 1991). From the theoretical side local spin models with AF coupling between nearest neighbor Fe moments in the Fe-Ni alloys have been used with success in explaining trends in the whole phase diagram by (Taylor and Gyorffy, 1992) as well as the thermal expansion anomaly and the increasing broadening of the specific heat around T c when approaching the critical composition Fe 65 Ni 35 (Gruner, Meyer and Entel, 1998). Local models may also be used to discuss the anti-invar effect (Gruner and Entel, 1998). With respect to the change of the anharmonicity in the Fe-Ni alloys upon mixing of FM and AF clusters it was already shown on the basis of full potential total-energy calculations that the AF solutions tend to have larger anharmonicities than the FM ones (Herper, Hoffmann and Entel, 1999), (Entel, Herper, Hoffmann, Nepecks et al., 2000). On the basis of calculations using the Korringa-Kohn-Rostocker Coherent-Potential Approximation (KKR-CPA), we have previously defined the Invar range of compositions by the existence of a NM or low-moment (LM) solution with small volume and a FM or high-moment (HM) solution with large volume. For Fe x Ni 100 x alloys in the fcc structure these solutions exist for 50 < x < 100. The recent calculations for Fe-Ni alloys show, that mainly due to AF-like contributions, 2

3 additional softening of the lattice occurs which explains the elastic anomalies of the Invar alloys at very low temperatures (van Schilfgaarde, Abrikosov and Johansson, 1999). However, the underlying physics could be more subtle. Early Mössbauer spectroscopy on the Fe-Ni samples from meteorites reveals the existence of a superstructure, i.e. Fe 65 Ni 35 decomposes into disordered fcc Fe 75 Ni 25 and an ordered L1 0 superstructure FeNi (Petersen, Aydin and Knudsen, 1977; Wiedemann, Li, Wagner and Petry, 1992). With respect to Fe-Pt alloys EXAFS studies on the local structure did not show essential difference between the ordered and disordered alloys, but revealed an unusual expansion of the Fe-Pt interatomic distance below the Curie temperature (Maruyama, Shirai, Maeda, Wan-Li et al., 1987). 2. COMPUTATIONAL RESULTS In order to check the importance of ASRO effects in the disordered alloys in the fcc structure we introduced the ASRO parameter S for Fe x Ni 1 x as well as for Fe-Pt alloys by c I Fe =(1 S)x/100, ci Ni =1 ci Fe (1) c II Fe =(1+S/3)x/100, cii Ni =1 cii Fe (2) where I and II refers to the corner and the face-centered site of the unit cell, respectively. The parameter S allows in particular to go continuously from disordered to ordered compounds. Positive values of S show increased Ni concentration at site I with respect to the average value, while negative values of S describe increased concentration of Fe at site I. Calculations were done by using the KKR-CPA method which allows different sites to have different chemical compositions. The calculations show that all compounds have two kind of states: One located at large volume, with ferromagnetic structure, and the other at lower volume with ferrimagnetic ordering. The equilibrium total energy variations for ferrimagnetic solutions as function of S for different compositions are shown in Fig. 1. All compounds show a decrease of the equilibrium total energy with increasing S. The equilibrium total energy variation for negative S has the same trend. The common behavior, for both positive and negative values of S, isthatlocal compositional variations lead to increasing stability of the alloy as well as peculiar behavior as a function of temperature and/or pressure. From the total-energy calculations for different S it is not clear which state corresponds on the average to the experimental situation. For example, choosing S =0.6for Fe 66 Ni 34 we obtain at site I and II the following concentrations for Fe and Ni: c I Fe = (for spin down), c II Fe = (for spin up), c I Ni = (for spin down), cii Ni = (for spin up), which leads to an alternating sequence of AF and FM layers. The AF-like layer prefers 3

4 a smaller lattice constant as compared to the FM layer, of which the competition leads to a broader total-energy curve as compared to the case of complete disorder (FM solution for S = 0). We take this as a proof that ASRO is necessary to obtain the additional negative lattice anharmonicity. This argument is further supported by the FLAPW calculation of (Herper, Hoffmann and Entel, 1999), in which case the competition between two different lattice constants is maximal. In order to be able to give a rough estimate of the average value to be chosen for S for a particular composition, we have calculated the bulk modulus B and the pressure derivative B by using a fit to Murnaghan s equation of state. The results for Fe 66 Ni 34 and Fe 65 Ni 35 are shown in Fig. 2. Experimentally B is small with values around zero for Invar compositions (Manosa, Saunders and Rahdi, 1992) which means that S =0.3 would be good choice for both Fe 66 Ni 34 and Fe 65 Ni 35.Wealsohave done calculations for Fe-Ni-Co super-invar alloys. They behave in much the same way as the Fe-Ni alloys. In case of Fe 72 Pt 28 the FM solution is always lower in energy compared to solutions with admixture of AF components. We believe that this is due to the larger lattice spacing in the Fe-Pt alloys. For S = 1 we do not find metastable AF solutions. The ordered alloy shows again a large magnetovolume effect. This result differs a bit from the result obtained by (Uhl, Sandratskii and Kübler, 1994). However, our calculations are salar relativistic, whereas Uhl at al. have performed non-relativistic calculations which might shift the energies of competing solutions. The curvature of the total energy curves as a function of the lattice constant is larger for the Fe-Pt alloys compared to the Fe-Ni alloys. The calculated values of B for Fe 72 Pt 28 are for all values of S very small and positive (the negative values of B in the experiments have been obtained only at room temperature (Manosa, Saunders, Rahdi, Kawald et al., 1991; Manosa, Saunders and Rahdi, 1992). To summarize we have investigated the influence of ASRO effects in FeNi, Fe-Ni-Co and Fe-Pt alloys. Although in all calculations the single-site CPA has been used (which means that we have neglected the small intersite T ij T -matrix elements), the allowance for inequivalent lattice sites I and II in the unit cell permits to obtain an impression of the importance of ordering tendencies in the alloys. For the Fe-Ni alloys ASRO leads to the appearance of AF components, and in general to NC moments, which simulates an alloy with an admixture of a small (AF) and a large (FM) lattice constant resulting in larger negative lattice anharmonicity. The microscopic origin of this is related to the behavior of the interatomic exchange integral (Sabiryanov and Bose, 1995), which leads to competing FM and AF interactions in the Fe-Ni alloys while this is less important in the Fe-Pt alloys due to the larger lattice constant. This competition is stabilized at low temperatures by a larger majority-spin DOS of antibonding t 2g states at E F leading to a smearing of the magnetovolume instability for some compositions, which, however, is still existent in Fe 72 Pt 28. Finally we would like to mention that we discussed ASRO effects in the Invar alloys only within the one-electron picture. Fur- 4

5 ther calculations, for example, of the atomic correlation functions by making use of the ab initio mean field method, (Johnson, Pinski, Staunton, Gyrffy et al., 1990; Johnson and Shelton, 1997), shed further light onto the finitetemperature behavior of the Invar alloys and is discussed in detail in (Crisan, Entel, Ebert, Akai et al.). Acknowledgements The authors acknowledge financial support by the DFG-Sonderforschungsbereich Nanopartikel aus der Gasphase: Entstehung, Struktur, Eigenschaften. References Crisan, V., P. Entel, H. Ebert, H. Akai et al. (). Magneto-chemical origin for Invar anomalies in Fe-(Ni, Co, Pt) alloys. Phys. Rev. B, submitted. Dubinin, S. F., S. K. Sidorov, S. G. Teploukhov and V. E. Arkhipov (1973). Coexistence of ferro- and antiferromagnetic order in Invar iron-nickel alloys. Soviet Phys. JETP Lett., 18, 324. Entel, P., H. C. Herper, E. Hoffmann, G. Nepecks et al. (2000). Understanding iron and its alloys from first principles. Phil.Mag.B,80, 141. Gruner, M. E. and P. Entel (1998). Monte Carlo simulations of magnetovolume instabilities in anti-invar systems. Comput. Mater. Sci., 10, 230. Gruner, M. E., R. Meyer and P. Entel (1998). Monte Carlo simulations of highmoment low-moment transitions in Invar alloys. Eur. Phys. J. B, 2, 107. Herper, H. C., E. Hoffmann and P. Entel (1999). Ab initio full-potential study of the structural and magnetic phase stability of iron. Phys. Rev. B, 60, Johnson, D. D., F. J. Pinski, J. B. Staunton, B. L. Gyrffy et al. (1990). NiFe Invar alloy: Theoretical insigts into the underlying mechanisms reponsible for their physical properties. In K. C. Russell and D. F. Smith, editors, Physical Metallurgy of Controlled Expansion Invar-Type Alloys, 3. TMS and Warrendale PA. Johnson, D. D. and W. A. Shelton (1997). Atomic Long- and Short-Range Order in Ni-Fe Invar Alloys. In J. Wittenauer, editor, The Invar Effect: A Centennial Symposium, 63. TMS and Warrendale, PA. Manosa, L., G. Saunders and H. Rahdi (1992). Acoustic-mode vibrational anharmonicity related to the anomalous thermal expansion of Invar iron alloys. Phys. Rev. B, 45, Manosa, L., G. A. Saunders, H. Rahdi, U. Kawald et al. (1991). Longitudinal acoustic mode softening and Invar behaviour in Fe 72,Pt 28. J. Phys.: Condens. Matter, 3,

6 Maruyama, H., K. Shirai, H. Maeda, L. Wan-Li et al. (1987). EXAFS studies of the local structure of phase transition in Fe-Pt Invar alloys. J. Phys. Soc. Japan, 56, Mohn, P. (1999). A century of zero expansion. Nature, 400, 18. Petersen, J. F., M. Aydin and J. M. Knudsen (1977). Mössbauer spectroscopy of an ordered phase (superstructure) of FeNi in an iron meteorite. Phys. Lett., 62A, 192. Sabiryanov, R. F. and S. K. Bose (1995). Effect of topological disorder on the itinerant magnetism of Fe and Co. Phys. Rev. B, 51, Schneider, T., M. Acet, E. F. Wassermann and W. Pepperhoff (1995). Antiferromagnetic Invar and anti-invar in Fe-Mn alloys. Phys. Rev. B, 51, Taylor, M. B. and B. L. Gyorffy (1992). Monte Carlo simulations of an FCC Ni c Fe 1 c alloy with vector magnetic freedom. J. Magn. Magn. Mater., , 877. Uhl, M., L. M. Sandratskii and J. Kübler (1994). Fluctuations in γ-fe and in Fe 3 Pt Invar from local-density-functional calculations. Phys. Rev. B, 50, 291. van Schilfgaarde, M., I. A. Abrikosov and B. Johansson (1999). Origin of the Invar effect in iron-nickel alloys. Nature, 400, 46. Wassermann, E. F. (1990). Moment-Volume instabilities in transition metals and alloys. In K. H. J. Buschow and E. P. Wohlfarth, editors, Ferromagnetic Materials, 240. North-Holland, Amsterdam. Wassermann, E. F. (1991). The Invar problem. J. Magn. Magn. Mater., 100, 346. Weiss, R. J. (1963). The origin of the Invar effect. Proc. Phys. Soc. London, 82, 281. Wiedemann, A., Q. Li, W. Wagner and W. Petry (1992). Fractal aggregation in Fe-Ni alloys during high temperature annealing. Physica B, 180 & 181, 793. Zähres, H., M. Acet, W. Stamm and E. F. Wassermann (1988). Coexisting antiferromagnetism and ferromagnetism in Fe-Ni Invar. J. Magn. Magn. Mater., 72, 80. 6

7 2 0 E (mry/cell) Fe 66 Ni 34 Fe 62 Ni 38 Fe 72 Pt 28 8 Fe 64.2 Ni 34 Co S Figure 1: Total energy variation, E = E(S) E(S = 0), as a function of the ASRO parameter S. The isolated symbols mark S = 0 FM solutions. 7

8 B, Fe 66 Ni 34 B, Fe 65 Ni 35 B, Fe 66 Ni 34 B, Fe 65 Ni B (GPa) B S 20 Figure 2: The bulk modulus and the pressure derivative of the bulk modulus for Fe 66 Ni 34 and Fe 65 Ni 35 Experiment and theory (B, S = 0): Open and filled diamond for Fe 64 Ni 36 and Fe 64.7 Ni 35.3 ; Experiment (B, S = 0): Open triangle down and star for Fe 58 Ni 42 and Fe 64 Ni 36 ; open triangles left and up for Fe 64.7 Ni 35.3 and Fe 65 Ni 35. Data are from Manosa et al. (1992) and Ref. therein. 8