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Two-ion magnetostriction in CoNiSiB amorphous samples
Journal of Magnetism and Magnetic Materials 148 (1995) L378-LA80
Letter to the Editor
Two-ion magnetostriction in CoNiSiB amorphous samples M. Magdoii aY *, A.Z. Maksymowicz
b, K. Kulakowski
b, J. Gonzilez
’
a Instituteof Physics and Computer Science, Pedagogical Vniuersity, ul. Podchorajch 2, 30-084 Cracow, Poland b Faculty of Physics and Nuclear Techniques, Vniuersiq of Mining and Metallurgy, Al. hickiewicza 30, 30-059 Cracow, Poland ’ Faculty of Chemistry, University of Basque Country, 20080 San Sebastian, Spain Received 13 February 1995; in revised form 16 May 1995
Abstract The contribution of anisotropic hopping to magnetostriction is calculated for a CoNiSiB amorphous sample. The calculations are performed for both as-quenched and annealed samples. Partial directional ordering of 3d atom pairs makes magnetostriction about twice smaller after the annealing, as was observed in experiment. Two-ion magnetic interaction in amorphous alloys is concluded to be responsible for annealing-induced changes of the magnetostriction AS.
1. Introduction In several experimental data it was found that magnetostriction coefficient A, measured at the same temperature was different after a thermal treatment of a sample. This result may be discussed in connection with possible structural changes. For example, in some CoFeSiB glasses different temperature dependence of A, was observed [l] for as-quenched and annealed samples. This was attributed to structural relaxations leading to a partial chemical short range order. In Ref. [2] the authors found a distinct change in concentration dependence of A,(x) in Pd,_,Fe, substitutional alloy in Fe-rich region. In amorphous Co,,Ni,,Si,,B,, alloy, magnetostriction h,(T) measured from helium to room temperature was significantly smaller after annealing while magnetization was nearly the same [3,4]. This result is in
* Corresponding author. [email protected]. 0304-8853/95/$09.50
Fax:
+ 48-12-372-243;
email:
contradiction to our previous calculations within the one-ion crystal-field magnetoelastic coupling model [5]; there, magnetostriction A, was found to increase when local strains are released. Such a reduction of strains is expected to occur when annealing is applied. In several experiments samples were annealed in a magnetic field. For example in [3,4] an electric current annealing with I = 4 A was applied during 3 h in air. Interaction between the effective magnetic field and the local easy axis of the atom pair may be the source of magnetic anisotropy of a sample, including anisotropy of magnetostriction. This suggests taking into account the distribution of atom pairs and their different directions. It is generally accepted [6,7] that topological or chemical ordering, or directional ordering of pairs may be the cause of changes of magnetic properties of amorphous systems. Such effects can be discussed within two-ion models where short range order or directional ordering are well defined. In this Letter we calculate the contribution to magnetostriction
0 1995 Elsevier Science B.V. All rights reserved
SSDI 0304-8853(95)00084-4
M. Magdoh et al. /Journal
caused by anisotropic nection with possible
of Magnetism and Magnetic Materials 148 (1995) L378-L380
hopping between ions, in constructural ordering.
2. Model Our theoretical model of an amorphous alloy is a set of independent pairs of atoms embedded in the ferromagnetic matrix. Two kinds of atoms are distinguished: magnetic (A) and nonmagnetic (II). For amorphous alloys such as Co,,Ni,,Si,,B,, both metals Co and Ni are counted as atoms A and the two left Si and B as atoms B. So there are three kinds of atom pairs: A-A, A-B and B-B. Every pair of atom can be orientated in seven chosen directions (edges and principal diagonals of a cube) to imitate the distribution of the pair directions. The directional ordering of atom pairs is described by the parameter q, one-axis anisotropy of pair directions, and in the lowest approximation we assume that p a exp[ q sin2/3 1, where p is the probability of finding a pair of atoms in the direction p. The parameter q determines the probabilities of atom pairs parallel to a given direction. The model Hamiltonian for a pair of atoms is H=
ME,,
.iivvi+ ~Ao,,(i)+i,
vui
i
- ~p~H;(2&+aiii) i upaij where the first term represents atomic levels depending on what atom occupies site i. The one-electron spin-orbit coupling is Aorb( Zeeman spin and orbital contributions with partial effective quenching (for (ui) of the orbital magnetic moments in metals are represented by the third term. Hopping integrals t,,(i,j) depend strongly on the distance between nearest neighbours (i,j). They are considered to be the source of the discussed changes of magnetostriction. These integrals depend on the angle of an atomic pair 181, giving rise to the anisotropic contribution. The Coulomb interaction treated within the Hartree-Fock approximation is included in the effective field. The Hamiltonian is diagonalized and each energy
L379
level is widened to account for the 3d band system. A more detailed description of the simplest version of the model may be found in Ref. [9]. Here we generalize the approach of Refs. [9,10] by introducing several possible directions for atom pairs and by taking into account all 2 X 2 X 3 = 12 states (two atoms X two spins X three orbitals). The present version of this approach was applied for the first time in [ll] to investigate the magnetization dependence of A, in homogeneous systems (only magnetic atoms). Local anisotropies are averaged over; however, long range orientational ordering of pairs can influence magnetic anisotropy of whole magnetic domains. This kind of averaging was discussed in Ref. [12] in terms of long-range strains. The model presented here tests for a possible contribution of anisotropic hopping to magnetostriction of systems with magnetic atom pairs embedded into a band. The amorphous state is imitated by appropriate averaging over different possible pair directions and different atomic (A-A, A-B and B-B) pairs in binary alloys. The diagonalization is performed numerically. Although we distinguish only between magnetic atoms (A) and nonmagnetic (B) ones in the amorphous Co,,Ni,,Si,,B,, alloy, their concentrations are taken into account in the whole numerical calculation, in particular to estimate the hopping integrals. Parameters used for numerical calculations are chosen so as to correspond to pairs of magnetic atoms with the spin-orbit coupling Aorb = 0.03 eV and effective hopping integrals of about 0.5 eV. The magnetic and nonmagnetic subbands are assumed to be of comparable widths. For numerical calculations we take into account only 3d states with t,, symmetry. The effective number of electrons is used as a fitting parameter for the magnetostriction versus temperature, with magnetic band split controlled by the molecular field of known magnetization.
3. Results and discussion The best fit is obtained for an effective number of electrons 0.268 of the maximum number allowed. This number is merely a fitting parameter, not the number of electrons in the d-band. This is so since we take into account only 3d states with r22gsymmetry.
M. Magdoh et al. /Journal
L380
0.00
100.00
Temperature
200.00 (normal
of Magnetism and Magnetic Materials 148 (1995) L378-L380
400.00 300.00 (red to Tc=320K)
Fig. 1. Temperature dependence of magnetostriction for amorphous Co,,Ni,,,Si,,B,,, alloy. Upper curves are for as quenched samples, lower ones are for after annealing (----experimental data after [3,4]; - - - calculated values.
calculations [9,10]. The obtained dependence is in a surprisingly good agreement with experiment (see Fig. 1). We conclude that anisotropic hopping may as well be a candidate for a two-ion contribution to magnetostriction and should not be excluded from a formal treatment. Although we have no arguments to exclude one-ion coupling, we note that the two-ion mechanism seems to be responsible for the influence of annealing to A,. We have not studied the topological disorder which also may greatly influence the magnetostriction. This will be taken into account in future calculations. It is necessary to remark that the temperature dependence of magnetostriction is very sensitive to details of the electronic structure, so our results of the calculations are only qualitative.
Acknowledgements As we assume the same bandwidth for magnetic and nonmagnetic components the resulting numbers are not sensitive with respect to chemical ordering. On the contrary, partial ordering of the pairs does influence magnetostriction and we are able to reproduce the experimentally observed tendency of about twice smaller magnetostriction after the annealing. Here we obtain a surplus of atom pairs in the plane normal to the z-axis along the linear stress; the obtained value of the anisotropy parameter q is - 0.7. We expect magnetostriction A, to be very sensitive to the details of the electronic structure, so only qualitative results may be obtained within our simple scheme. However, the results of our calculations are in good agreement with experimental data, as seen in Fig. 1.
4. Conclusions The contribution of anisotropic hopping to magnetostriction .of systems with magnetic atom pairs is calculated for a set of parameters for alloy and compared with experiCo,,Ni,,Si,,B,, mental data for as-quenched and annealed samples. The model approach is the generalization of previous
This work was supported by the Polish Committee for Scientific Research, grant No. 2 P03B 008 08.
References [I] J.M. Barandiaran, A. Hernando, V. Madurga, O.V. Nielsen, M. Vazquez and M. Vazquez-Lopez, Phys. Rev. B 35 (1987) 5066. [2] J.E. Schmidt and L. Berger, J. Appl. Phys. 55 (1984) 1073. [3] E. du Tremolet de Lacheisserie and J. Gonzalez, J. de Physique 50 (1989) 939. [4] J. Gonzalez and E. du Tremolet de Lacheisserie, Phys. Stat. Sol. (a) 115 (1989) K233. [5] K. Kulakowski, A. Maksymowicz and M. Magdoii, J. Magn. Magn. Mater. 115 (1992) L143. [6] R.C. O’Handley, J. Appl. Phys. 62 (1987) R15. [7] E. du Tremolet de Lacheisserie, Magnetostriction - Theory and Applications of Magnetoelasticity (CRC Press, Boca Raton, FL, 1993). [8] J.C. Slater and G.F. Koster, Phys. Rev. 94 (1954) 1498. [9] K. Ktdakowski, J. Gonzalez, J. Magn. Magn. Mater. 123 (1993) 169. [lo] J. Gonzalez, J.M. Blanco, PG. Barbon and K. Ktdakowski, J. Magn. Magn. Mater. 102 (1991) 63. [ll] K. Kulakowski, A. Maksymowicz and M. Magdoh, Acta Phys. Pol. A 85 (1994) 869. [12] J.M. Barandiaran and A. Hernando, J. Magn. Magn. Mater. 104-107 (1992) 73.
Letter to the Editor
Two-ion magnetostriction in CoNiSiB amorphous samples M. Magdoii aY *, A.Z. Maksymowicz
b, K. Kulakowski
b, J. Gonzilez
’
a Instituteof Physics and Computer Science, Pedagogical Vniuersity, ul. Podchorajch 2, 30-084 Cracow, Poland b Faculty of Physics and Nuclear Techniques, Vniuersiq of Mining and Metallurgy, Al. hickiewicza 30, 30-059 Cracow, Poland ’ Faculty of Chemistry, University of Basque Country, 20080 San Sebastian, Spain Received 13 February 1995; in revised form 16 May 1995
Abstract The contribution of anisotropic hopping to magnetostriction is calculated for a CoNiSiB amorphous sample. The calculations are performed for both as-quenched and annealed samples. Partial directional ordering of 3d atom pairs makes magnetostriction about twice smaller after the annealing, as was observed in experiment. Two-ion magnetic interaction in amorphous alloys is concluded to be responsible for annealing-induced changes of the magnetostriction AS.
1. Introduction In several experimental data it was found that magnetostriction coefficient A, measured at the same temperature was different after a thermal treatment of a sample. This result may be discussed in connection with possible structural changes. For example, in some CoFeSiB glasses different temperature dependence of A, was observed [l] for as-quenched and annealed samples. This was attributed to structural relaxations leading to a partial chemical short range order. In Ref. [2] the authors found a distinct change in concentration dependence of A,(x) in Pd,_,Fe, substitutional alloy in Fe-rich region. In amorphous Co,,Ni,,Si,,B,, alloy, magnetostriction h,(T) measured from helium to room temperature was significantly smaller after annealing while magnetization was nearly the same [3,4]. This result is in
* Corresponding author. [email protected]. 0304-8853/95/$09.50
Fax:
+ 48-12-372-243;
email:
contradiction to our previous calculations within the one-ion crystal-field magnetoelastic coupling model [5]; there, magnetostriction A, was found to increase when local strains are released. Such a reduction of strains is expected to occur when annealing is applied. In several experiments samples were annealed in a magnetic field. For example in [3,4] an electric current annealing with I = 4 A was applied during 3 h in air. Interaction between the effective magnetic field and the local easy axis of the atom pair may be the source of magnetic anisotropy of a sample, including anisotropy of magnetostriction. This suggests taking into account the distribution of atom pairs and their different directions. It is generally accepted [6,7] that topological or chemical ordering, or directional ordering of pairs may be the cause of changes of magnetic properties of amorphous systems. Such effects can be discussed within two-ion models where short range order or directional ordering are well defined. In this Letter we calculate the contribution to magnetostriction
0 1995 Elsevier Science B.V. All rights reserved
SSDI 0304-8853(95)00084-4
M. Magdoh et al. /Journal
caused by anisotropic nection with possible
of Magnetism and Magnetic Materials 148 (1995) L378-L380
hopping between ions, in constructural ordering.
2. Model Our theoretical model of an amorphous alloy is a set of independent pairs of atoms embedded in the ferromagnetic matrix. Two kinds of atoms are distinguished: magnetic (A) and nonmagnetic (II). For amorphous alloys such as Co,,Ni,,Si,,B,, both metals Co and Ni are counted as atoms A and the two left Si and B as atoms B. So there are three kinds of atom pairs: A-A, A-B and B-B. Every pair of atom can be orientated in seven chosen directions (edges and principal diagonals of a cube) to imitate the distribution of the pair directions. The directional ordering of atom pairs is described by the parameter q, one-axis anisotropy of pair directions, and in the lowest approximation we assume that p a exp[ q sin2/3 1, where p is the probability of finding a pair of atoms in the direction p. The parameter q determines the probabilities of atom pairs parallel to a given direction. The model Hamiltonian for a pair of atoms is H=
ME,,
.iivvi+ ~Ao,,(i)+i,
vui
i
- ~p~H;(2&+aiii) i upaij where the first term represents atomic levels depending on what atom occupies site i. The one-electron spin-orbit coupling is Aorb( Zeeman spin and orbital contributions with partial effective quenching (for (ui) of the orbital magnetic moments in metals are represented by the third term. Hopping integrals t,,(i,j) depend strongly on the distance between nearest neighbours (i,j). They are considered to be the source of the discussed changes of magnetostriction. These integrals depend on the angle of an atomic pair 181, giving rise to the anisotropic contribution. The Coulomb interaction treated within the Hartree-Fock approximation is included in the effective field. The Hamiltonian is diagonalized and each energy
L379
level is widened to account for the 3d band system. A more detailed description of the simplest version of the model may be found in Ref. [9]. Here we generalize the approach of Refs. [9,10] by introducing several possible directions for atom pairs and by taking into account all 2 X 2 X 3 = 12 states (two atoms X two spins X three orbitals). The present version of this approach was applied for the first time in [ll] to investigate the magnetization dependence of A, in homogeneous systems (only magnetic atoms). Local anisotropies are averaged over; however, long range orientational ordering of pairs can influence magnetic anisotropy of whole magnetic domains. This kind of averaging was discussed in Ref. [12] in terms of long-range strains. The model presented here tests for a possible contribution of anisotropic hopping to magnetostriction of systems with magnetic atom pairs embedded into a band. The amorphous state is imitated by appropriate averaging over different possible pair directions and different atomic (A-A, A-B and B-B) pairs in binary alloys. The diagonalization is performed numerically. Although we distinguish only between magnetic atoms (A) and nonmagnetic (B) ones in the amorphous Co,,Ni,,Si,,B,, alloy, their concentrations are taken into account in the whole numerical calculation, in particular to estimate the hopping integrals. Parameters used for numerical calculations are chosen so as to correspond to pairs of magnetic atoms with the spin-orbit coupling Aorb = 0.03 eV and effective hopping integrals of about 0.5 eV. The magnetic and nonmagnetic subbands are assumed to be of comparable widths. For numerical calculations we take into account only 3d states with t,, symmetry. The effective number of electrons is used as a fitting parameter for the magnetostriction versus temperature, with magnetic band split controlled by the molecular field of known magnetization.
3. Results and discussion The best fit is obtained for an effective number of electrons 0.268 of the maximum number allowed. This number is merely a fitting parameter, not the number of electrons in the d-band. This is so since we take into account only 3d states with r22gsymmetry.
M. Magdoh et al. /Journal
L380
0.00
100.00
Temperature
200.00 (normal
of Magnetism and Magnetic Materials 148 (1995) L378-L380
400.00 300.00 (red to Tc=320K)
Fig. 1. Temperature dependence of magnetostriction for amorphous Co,,Ni,,,Si,,B,,, alloy. Upper curves are for as quenched samples, lower ones are for after annealing (----experimental data after [3,4]; - - - calculated values.
calculations [9,10]. The obtained dependence is in a surprisingly good agreement with experiment (see Fig. 1). We conclude that anisotropic hopping may as well be a candidate for a two-ion contribution to magnetostriction and should not be excluded from a formal treatment. Although we have no arguments to exclude one-ion coupling, we note that the two-ion mechanism seems to be responsible for the influence of annealing to A,. We have not studied the topological disorder which also may greatly influence the magnetostriction. This will be taken into account in future calculations. It is necessary to remark that the temperature dependence of magnetostriction is very sensitive to details of the electronic structure, so our results of the calculations are only qualitative.
Acknowledgements As we assume the same bandwidth for magnetic and nonmagnetic components the resulting numbers are not sensitive with respect to chemical ordering. On the contrary, partial ordering of the pairs does influence magnetostriction and we are able to reproduce the experimentally observed tendency of about twice smaller magnetostriction after the annealing. Here we obtain a surplus of atom pairs in the plane normal to the z-axis along the linear stress; the obtained value of the anisotropy parameter q is - 0.7. We expect magnetostriction A, to be very sensitive to the details of the electronic structure, so only qualitative results may be obtained within our simple scheme. However, the results of our calculations are in good agreement with experimental data, as seen in Fig. 1.
4. Conclusions The contribution of anisotropic hopping to magnetostriction .of systems with magnetic atom pairs is calculated for a set of parameters for alloy and compared with experiCo,,Ni,,Si,,B,, mental data for as-quenched and annealed samples. The model approach is the generalization of previous
This work was supported by the Polish Committee for Scientific Research, grant No. 2 P03B 008 08.
References [I] J.M. Barandiaran, A. Hernando, V. Madurga, O.V. Nielsen, M. Vazquez and M. Vazquez-Lopez, Phys. Rev. B 35 (1987) 5066. [2] J.E. Schmidt and L. Berger, J. Appl. Phys. 55 (1984) 1073. [3] E. du Tremolet de Lacheisserie and J. Gonzalez, J. de Physique 50 (1989) 939. [4] J. Gonzalez and E. du Tremolet de Lacheisserie, Phys. Stat. Sol. (a) 115 (1989) K233. [5] K. Kulakowski, A. Maksymowicz and M. Magdoii, J. Magn. Magn. Mater. 115 (1992) L143. [6] R.C. O’Handley, J. Appl. Phys. 62 (1987) R15. [7] E. du Tremolet de Lacheisserie, Magnetostriction - Theory and Applications of Magnetoelasticity (CRC Press, Boca Raton, FL, 1993). [8] J.C. Slater and G.F. Koster, Phys. Rev. 94 (1954) 1498. [9] K. Ktdakowski, J. Gonzalez, J. Magn. Magn. Mater. 123 (1993) 169. [lo] J. Gonzalez, J.M. Blanco, PG. Barbon and K. Ktdakowski, J. Magn. Magn. Mater. 102 (1991) 63. [ll] K. Kulakowski, A. Maksymowicz and M. Magdoh, Acta Phys. Pol. A 85 (1994) 869. [12] J.M. Barandiaran and A. Hernando, J. Magn. Magn. Mater. 104-107 (1992) 73.