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Effect of atomic ordering on linear magneto striction in PdFe alloys
Journal of Magnetism and Magnetic Materials 160 (1996) 53-54
~a ELSEVIER
Journalof magnetism and magnetic materials
Effect of atomic ordering on linear magnetostriction in PdFe alloys A.Z. Maksymowicz a,*, M.S. Magdofi b a Faculo' of Physics and Nuclear Techniques AGH, Al. Mickiewicza 30, 30-059 Krakdw, Poland b Institute of Physics and Computer Science, Pedagogical University UI. Podchorq~ych 2, 30-084 Krakdw, Poland Abstract A simplified band model calculation of magnetostriction for Pd~ xFex substitutional alloys was carried out to observe the effect of atomic order on magnetostriction. It is suggested that ordering through quenching and annealing may influence magnetostriction. Different anisotropic electronic hopping integrals between possible atomic pairs of the alloy may be considered a possible source of that dependence. We obtained reasonable agreement with experiment.
Keywords: Alloys; Atomic ordering; Magnetostriction
1. Introduction Different concentration dependences of magnetostrictlon have been observed in, for example, Pd 1_,Fe~ substitutional alloys in the Fe-rich region after thermal treatment of samples [1]. It was suggested that the atomic ordering which takes place through quenching and annealing may be responsible for the observed changes. We consider a simplified model with anisotropic hopping as a possible source of the dependence of magnetostriction on chemical ordering. Different hopping of electrons between possible pairs in the alloy consisting of Pd and Fe atoms offers a sensitive response of the local environment to the magnetostriction. In this paper we calculate the contribution to magnetostriction from anisotropic hopping and account for the chemical (atomic) order induced by the annealing and therefore responsible for the observed changes in magnetostriction. 2. Model The model Hamiltonian for a pair of atoms is
H = ~E~t(i)~,,,, ' + Y'~ao~t,(i)6iL i VO'~
i
-- E/ZBHi(20", + aiLi) + E i
tvu(i,J)av+,a~o-,,
tJl.to-ij
(l)
* Corresponding author. Fax: +48-12-339-406.
where the first term represents the atomic level of an atom on site i. The one-electron spin-orbit coupling is Aorb(i). Zeeman spin and orbital contributions with partial effective quenching (for c~i va 0) of the orbital magnetic moments of metals are represented by the third term. Hopping integrals t~u(i,j) depend strongly on the distance between nearest neighbours. These integrals depend on the angle of an atomic pair with respect to the crystallographic axis, giving rise to the anisotropic contribution. The Hamiltonian with the hopping terms is treated in a simplified manner where the alloy is considered as a set of independent pairs of atoms embedded in the ferromagnetic matrix. The determined atomic levels are widened to simulate the effect of the effective medium. Coulomb interaction within the Hartree-Fock approximation is included in the effective field. The calculation proceeds in the following way. First, the contribution to magnetostriction of an atomic pair with arbitrary possible orientation is calculated by a microscopic model Hamiltonian. Magnetostriction enters into the calculations via the anisotropic dependence of the hopping integrals t on the distance R between nearest neighbours. In fact, we assumed the power low t(R) ~ 1 / R n, where n depends on the orbitals of the atoms between which the electron jumps. We take n = 2 for Pd-Pd, n = 3.5 for Pd-Fe and n = 5 for F e - F e pairs when hopping between d-states occurs. Next, a statistical averaging is performed over independent (i,j) pairs according to an assumed pair distribution function at given annealing temperature. A more detailed description of the simplest version of the model may be found in Ref. [2]. (It should be remarked that the statistical summation over the magnetostrictive
0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0304-885 3(96)00 108-4
A.Z Maksymowicz, M.S. Magdo/t /Journal (?['Magnetism and Magnetic Materials 160 (1996) 53-54
54
strains of independent atomic pairs is a rough approximation, because in reality the magneto-elastic incompatibilities should be taken into account, as was discussed in Ref. [3] and references therein.) Parameters used for numerical calculations are chosen so as to correspond to pairs of magnetic atoms with s p i n - o r b i t coupling Aorb = 0.03 eV and effective hopping integrals o f about 0.5 eV, which may be assumed to be roughly the same for all transition metals. The quenching parameter a = 0.05 was used. The magnetic and nonmagnetic sub-bands are assumed to be equal to the bandwidth of pure Fe and pure Pd, respectively. For numerical calculations we only take into account 3d states with t2~ symmetry. The input number of electrons is estimated from the assumption o f strong magnetism on Fe sites for iron concentrations below 68% [1]. Then the saturation magnetization, needed for the number of electrons on iron sites, was taken from the first-principles calculation o f Ref. [4]. For iron concentrations above 68%, we assumed a linear dependence o f 3d electrons with slope chosen so as to obtain the c o m m o n l y assumed number of 7.18 electrons for pure iron. The calculations were carried out for two limiting cases: a random alloy corresponding to asquenched samples, and a m a x i m u m allowed atomic ordered case (corresponding to an annealed sample). The number o f 3d electrons on iron sites is n = 10 - m,
(2)
for the strong magnetism case x < 0.68, where m is the calculated magnetization [4] on iron sites. Then n proved to be a linear function o f x with a slope which extrapolates to 7.18 for the number o f electrons for pure Fe, a very probable value. It follows from Ref. [4] that re(x) is also a linear function for 0.68 _< x _< 1, with m = 3.01 at x = 0.68 and m = 2.22 for iron, i.e. x = 1. 3.
Results
and
discussion
The main results are collected in Table 1. It can be seen that atomic ordering has little effect in the region where the saturated magnetization is o f a strong type, e.g. when one o f the magnetic sub-bands is fully occupied. In the Fe-rich region, however, a change in the occupation number o f electrons in the two magnetic sub-bands takes place when the concentration (and magnetization) changes. The system, which is rather sensitive to the number o f electrons [5], then responds more rapidly, resulting in a more profound change of the magnetostriction versus the local environment which is described by the atomic order. The agreement with experiment is reasonable. W e conclude that anisotropic hopping is a candidate for the atomic order dependence o f magnetostriction in weak magnetics, e.g. for the lower range o f Pd concentration up to about 20% in Pd t .~Fe ~..
Table 1 Concentration (r), assumed number of 3d electrons on Fe sites (n) and magnetization m (in Bohr magneton, p, BY Experimental and calculated values of magnetostriction Aq for quenched (Qnd.) and annealed (ann.) samples (in arbitrary units) m
Aq experiment
Aq calculated
(/z B)
Qnd.
Ann.
Qnd.
Ann.
0
0
0
0
x
n
0.00 0.01 0.02 0.03 0.05 0.08 0.10 0.12 0.14 0.15 0.16 0.18 0.20 0.25 0.30 0.33 0.35 0.36 0.40 0.50 0.60 0.65 0.68 0.70 0.75 0.80 0.85 0.90 0.92 0.95 0.98 0.99 1.00
6.58 6.59 6.59
3.42 3.41 3.41
6.61
3.39
6.64
3.36
-
1.9
-
0.5 1.4
6.67
2
3
3
3
3
0.6 -7.1 -6.6 1.3 10.0 22.5 29.8
-6.9 -6.0 -2.1 13.1 28.2 33.5
2
1
-2 -2 0
-2 -3 - 1
30.9 32.2 49.6 77.2
34.8
30
27
46 58 59 57 55 54 45 35 21 l0 5 0 -11 -15
42 56 57 54 49 42 22 4 -13 -22 22 -21 -21 -20 - 19
51.8
15.4
4.9
-3.3
41.9
-6.8
3.33
6.70 6.73 6.76
3.30 3.27 3.24
6.79
3.2l
6.82 6.88 6.94 6.97 6.99 7.00 7.03 7.06 7.09 7.12 7.13 7.15 7.17 7.17 7.18
1
2 1,9 0.5 2.2 4.4 2.0
1.5
1
3.18 3.12 3.06 3.03 3.01 2.96 2.84 2.71 2.59 2.47 2.42 2.34 2.27 2.24 2.22
-
13.2
-
13.2
-
19
Acknowledgement: This work was partly supported by KBN grant 2 P03B 008 08. References
[1] J.E. Schmidt and L. Berger, J. Appl. Phys. 55 (1984) 1073. [2] K. Kutakowski and J. Gonzgdez, J. Magn. Magn. Mater. 123 (1993) 169. [3] M. F~ihnIe, J. Furthmfiller, R. Pawellek and T. Beuerle, Appl. Phys. Lett. 59 (1991) 2049. [4] A.Z. Maksymowicz, S. Kaprzyk and K. Zakrzewska, Phys. Rev. B, accepted (1996). [5] M. Magdofi. A.Z. Maksymowicz, K. Kulakowski and J. Gonz~.lez, J. Magn. Magn. Mater. 148 (1995) L378.
~a ELSEVIER
Journalof magnetism and magnetic materials
Effect of atomic ordering on linear magnetostriction in PdFe alloys A.Z. Maksymowicz a,*, M.S. Magdofi b a Faculo' of Physics and Nuclear Techniques AGH, Al. Mickiewicza 30, 30-059 Krakdw, Poland b Institute of Physics and Computer Science, Pedagogical University UI. Podchorq~ych 2, 30-084 Krakdw, Poland Abstract A simplified band model calculation of magnetostriction for Pd~ xFex substitutional alloys was carried out to observe the effect of atomic order on magnetostriction. It is suggested that ordering through quenching and annealing may influence magnetostriction. Different anisotropic electronic hopping integrals between possible atomic pairs of the alloy may be considered a possible source of that dependence. We obtained reasonable agreement with experiment.
Keywords: Alloys; Atomic ordering; Magnetostriction
1. Introduction Different concentration dependences of magnetostrictlon have been observed in, for example, Pd 1_,Fe~ substitutional alloys in the Fe-rich region after thermal treatment of samples [1]. It was suggested that the atomic ordering which takes place through quenching and annealing may be responsible for the observed changes. We consider a simplified model with anisotropic hopping as a possible source of the dependence of magnetostriction on chemical ordering. Different hopping of electrons between possible pairs in the alloy consisting of Pd and Fe atoms offers a sensitive response of the local environment to the magnetostriction. In this paper we calculate the contribution to magnetostriction from anisotropic hopping and account for the chemical (atomic) order induced by the annealing and therefore responsible for the observed changes in magnetostriction. 2. Model The model Hamiltonian for a pair of atoms is
H = ~E~t(i)~,,,, ' + Y'~ao~t,(i)6iL i VO'~
i
-- E/ZBHi(20", + aiLi) + E i
tvu(i,J)av+,a~o-,,
tJl.to-ij
(l)
* Corresponding author. Fax: +48-12-339-406.
where the first term represents the atomic level of an atom on site i. The one-electron spin-orbit coupling is Aorb(i). Zeeman spin and orbital contributions with partial effective quenching (for c~i va 0) of the orbital magnetic moments of metals are represented by the third term. Hopping integrals t~u(i,j) depend strongly on the distance between nearest neighbours. These integrals depend on the angle of an atomic pair with respect to the crystallographic axis, giving rise to the anisotropic contribution. The Hamiltonian with the hopping terms is treated in a simplified manner where the alloy is considered as a set of independent pairs of atoms embedded in the ferromagnetic matrix. The determined atomic levels are widened to simulate the effect of the effective medium. Coulomb interaction within the Hartree-Fock approximation is included in the effective field. The calculation proceeds in the following way. First, the contribution to magnetostriction of an atomic pair with arbitrary possible orientation is calculated by a microscopic model Hamiltonian. Magnetostriction enters into the calculations via the anisotropic dependence of the hopping integrals t on the distance R between nearest neighbours. In fact, we assumed the power low t(R) ~ 1 / R n, where n depends on the orbitals of the atoms between which the electron jumps. We take n = 2 for Pd-Pd, n = 3.5 for Pd-Fe and n = 5 for F e - F e pairs when hopping between d-states occurs. Next, a statistical averaging is performed over independent (i,j) pairs according to an assumed pair distribution function at given annealing temperature. A more detailed description of the simplest version of the model may be found in Ref. [2]. (It should be remarked that the statistical summation over the magnetostrictive
0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0304-885 3(96)00 108-4
A.Z Maksymowicz, M.S. Magdo/t /Journal (?['Magnetism and Magnetic Materials 160 (1996) 53-54
54
strains of independent atomic pairs is a rough approximation, because in reality the magneto-elastic incompatibilities should be taken into account, as was discussed in Ref. [3] and references therein.) Parameters used for numerical calculations are chosen so as to correspond to pairs of magnetic atoms with s p i n - o r b i t coupling Aorb = 0.03 eV and effective hopping integrals o f about 0.5 eV, which may be assumed to be roughly the same for all transition metals. The quenching parameter a = 0.05 was used. The magnetic and nonmagnetic sub-bands are assumed to be equal to the bandwidth of pure Fe and pure Pd, respectively. For numerical calculations we only take into account 3d states with t2~ symmetry. The input number of electrons is estimated from the assumption o f strong magnetism on Fe sites for iron concentrations below 68% [1]. Then the saturation magnetization, needed for the number of electrons on iron sites, was taken from the first-principles calculation o f Ref. [4]. For iron concentrations above 68%, we assumed a linear dependence o f 3d electrons with slope chosen so as to obtain the c o m m o n l y assumed number of 7.18 electrons for pure iron. The calculations were carried out for two limiting cases: a random alloy corresponding to asquenched samples, and a m a x i m u m allowed atomic ordered case (corresponding to an annealed sample). The number o f 3d electrons on iron sites is n = 10 - m,
(2)
for the strong magnetism case x < 0.68, where m is the calculated magnetization [4] on iron sites. Then n proved to be a linear function o f x with a slope which extrapolates to 7.18 for the number o f electrons for pure Fe, a very probable value. It follows from Ref. [4] that re(x) is also a linear function for 0.68 _< x _< 1, with m = 3.01 at x = 0.68 and m = 2.22 for iron, i.e. x = 1. 3.
Results
and
discussion
The main results are collected in Table 1. It can be seen that atomic ordering has little effect in the region where the saturated magnetization is o f a strong type, e.g. when one o f the magnetic sub-bands is fully occupied. In the Fe-rich region, however, a change in the occupation number o f electrons in the two magnetic sub-bands takes place when the concentration (and magnetization) changes. The system, which is rather sensitive to the number o f electrons [5], then responds more rapidly, resulting in a more profound change of the magnetostriction versus the local environment which is described by the atomic order. The agreement with experiment is reasonable. W e conclude that anisotropic hopping is a candidate for the atomic order dependence o f magnetostriction in weak magnetics, e.g. for the lower range o f Pd concentration up to about 20% in Pd t .~Fe ~..
Table 1 Concentration (r), assumed number of 3d electrons on Fe sites (n) and magnetization m (in Bohr magneton, p, BY Experimental and calculated values of magnetostriction Aq for quenched (Qnd.) and annealed (ann.) samples (in arbitrary units) m
Aq experiment
Aq calculated
(/z B)
Qnd.
Ann.
Qnd.
Ann.
0
0
0
0
x
n
0.00 0.01 0.02 0.03 0.05 0.08 0.10 0.12 0.14 0.15 0.16 0.18 0.20 0.25 0.30 0.33 0.35 0.36 0.40 0.50 0.60 0.65 0.68 0.70 0.75 0.80 0.85 0.90 0.92 0.95 0.98 0.99 1.00
6.58 6.59 6.59
3.42 3.41 3.41
6.61
3.39
6.64
3.36
-
1.9
-
0.5 1.4
6.67
2
3
3
3
3
0.6 -7.1 -6.6 1.3 10.0 22.5 29.8
-6.9 -6.0 -2.1 13.1 28.2 33.5
2
1
-2 -2 0
-2 -3 - 1
30.9 32.2 49.6 77.2
34.8
30
27
46 58 59 57 55 54 45 35 21 l0 5 0 -11 -15
42 56 57 54 49 42 22 4 -13 -22 22 -21 -21 -20 - 19
51.8
15.4
4.9
-3.3
41.9
-6.8
3.33
6.70 6.73 6.76
3.30 3.27 3.24
6.79
3.2l
6.82 6.88 6.94 6.97 6.99 7.00 7.03 7.06 7.09 7.12 7.13 7.15 7.17 7.17 7.18
1
2 1,9 0.5 2.2 4.4 2.0
1.5
1
3.18 3.12 3.06 3.03 3.01 2.96 2.84 2.71 2.59 2.47 2.42 2.34 2.27 2.24 2.22
-
13.2
-
13.2
-
19
Acknowledgement: This work was partly supported by KBN grant 2 P03B 008 08. References
[1] J.E. Schmidt and L. Berger, J. Appl. Phys. 55 (1984) 1073. [2] K. Kutakowski and J. Gonzgdez, J. Magn. Magn. Mater. 123 (1993) 169. [3] M. F~ihnIe, J. Furthmfiller, R. Pawellek and T. Beuerle, Appl. Phys. Lett. 59 (1991) 2049. [4] A.Z. Maksymowicz, S. Kaprzyk and K. Zakrzewska, Phys. Rev. B, accepted (1996). [5] M. Magdofi. A.Z. Maksymowicz, K. Kulakowski and J. Gonz~.lez, J. Magn. Magn. Mater. 148 (1995) L378.