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Spin waves in thin films composed of ferromagnetic and nonmagnetic materials
Journal of Magnetismand MagneticMaterials 140-144 (1995) 1945-1946
•1•
~i ELSEVIER
Journalof magnetism and magnetic materials
Spin waves in thin films composed of ferromagnetic and nonmagnetic materials R. S w i r k o w i c z
*
Institute of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland Abstract Systems composed of Ni and nonmagnetic layers are studied within the framework of the multiband approach. Spin-wave spectra and the amplitudes of modes are investigated. Changes in the strength of the effective coupling between Ni layers are discussed.
I. Introduction
Recently, films composed of magnetic and nonmagnetic materials have been studied. Changes of thickness of a nonmagnetic spacer lead to changes in the strength of the magnetic coupling and substantially influence the spectrum of collective modes. The spin-wave formalism within the framework of the band model was used to determine the coupling between ferromagnetic monolayers embedded in nonmagnetic metal [1]. Problems of the dynamic susceptibility and spin waves in double layers were investigated [2]. In the present paper, structures composed of nickel and nonmagnetic transition metals are studied. 2. The model
Systems composed of Ni sublayers which are separated by k nonmagnetic planes are investigated. Calculations are performed within the framework of the multiband model with five d-bands taken into account. The tight-binding approximation with Slater-Koster parameterization is used. Parameters appropriate for Ni and nonmagnetic metals (Cu, Au, Ag) are taken. The one-electron spin-independent potential is chosen in such a way that the correct number of d-electrons in both metals is obtained. The Coulomb integral U equal to 1.0 eV is chosen in all layers. The surface of the film and the interfaces are perpendicular to the [001] direction. The band structure is calculated in the Hartree-Fock approximation and profiles of the ground-state magnetization are found. The magnetization is slightly enhanced in
Ni layers corresponding to the surfaces, but it is lowered at Ni-nonmagnetic metal interfaces. A very small magnetic moment is induced in the planes of nonmagnetic metals which are in direct contact with Ni. Spin waves are investigated within the framework of the Random Phase Approximation using the dynamic susceptibility method. The transverse susceptibility is calculated in the mixed Bloch-Wannier representation [1,2] for various values of a two-dimensional wave vector q parallel to the interfaces. 3. Spin waves. Results and discussion
Spin waves are investigated for systems which are composed of two Ni sublayers, each containing N = 5 atomic planes, and separated by k sheets of non-magnetic metal. The thickness of the spacer is varied from k = 1 to 7 atomic layers. In the spin-wave spectrum, pairs of modes with close energies are found. The differences in energy corresponding to two pairs of low-lying modes are depicted in Fig. 1 as a function of k. Calculations are performed for a Cu spacer and q = 0. The energy gap between two branches is a distinctive feature of a given pair and it diminishes with an increase in k. For k = 1 the
m
t'!
w
21D
k
* Fax: 48-2-628-21-71;email: [email protected].
o
"t'4"4" k
Fig. 1. Energy gaps for the first (a) and the second (b) pairs of low-lying modes in the case of a double layer versus k (q = 0).
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0304-8853(94)01561-9
1946
R. Swirkowicz /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1945-1946
~ ~ ~ ( 1 . 0 )
Fig. 2. Dispersion relations for a double layer with Ag spacer. gap corresponding to the second pair is considerably greater than for the first. The energy gap between the two lowest modes with q = 0 can be considered as a measure of the indirect exchange coupling between ferromagnetic films [1]. Therefore, in the systems under consideration the effective exchange coupling strongly diminishes when the thickness of the spacer increases from 1 to 7 atomic layers. Spin-wave dispersion relations are calculated for systems composed of two Ni sublayers separated by k = 3 spacer planes. The parameters are modified in such a way that the spacer corresponding to Cu, Au or Ag is chosen. The general course of the dispersion curves is similar in all cases. In the region of small q, pairs of modes with close energies are found (Fig. 2). The distance in energy for a given pair is greater for higher q. The amplitudes of a pair of modes are practically the same. In Ni planes which are in direct contact with the nonmagnetic spacer the amplitudes are relatively high, whereas in the spacer layers they are negligibly small. A considerable discontinuity appears near the boundary, which is well seen for antisymmetric modes. Similar calculations are performed for systems composed of 3 Ni sublayers. Dispersion curves obtained for low-lying modes are presented in Fig. 3 for the case of a Cu spacer and k = 1. The subband which consists of three
modes is well seen in the figure. Two of the modes in the group are mainly localized in the outer Ni sublayers, whereas the third one is essentially confined in the central part of the film. Because of damping effects, it is difficult to determine precisely the energies of high-lying modes. Only some points corresponding to the higher subbands are presented in the figure. Energies characteristic for spin waves in a separate Ni film consisting of N = 5 atomic planes are marked in the figure by bars. There is a consistency between values of energies in both systems. Therefore, the modes have a tendency to form groups consisting of several branches with energy values close to the ones characteristic for a separate film. Such results show that the effective magnetic interactions between Ni sublayers are weak. Next, calculations are performed for a system composed of 5 Ni sublayers separated by Cu spacers with k = 2. In the region of small q, the five lowest modes are practically degenerated. Two of the modes are localized in Ni sublayers close to the surfaces. Three other modes in the group have considerable amplitudes in inner Ni sublayers. In the spin-wave spectrum two subbands can be seen. Results presented in this paper are consistent with those obtained in Heisenberg and phenomenological approaches [3,4]. Pairs of modes with close energies appear to be a characteristic feature of the spin-wave spectrum of double films and are observed experimentally [3,4]. Relatively high amplitudes and a considerable discontinuity of antisymmetric modes at the interfaces are obtained in the Heisenberg model [3]. However, compared to the results of the phenomenological approach [4] it seems that the calculated effective coupling between magnetic layers diminishes too fast in the case of thicker interlayers. It is possible that in the systems under consideration, electrons of s symmetry play an important role and influence the effective coupling between magnetic sublayers for the thick spacer [5]. Acknowledgement: The work was supported by the Scientific Research Committee, project No 2 2321 9102.
References
10oi ,o)
Fig. 3. Dispersion relations for three Ni sublayers with Cu spacer.
[1] M.S. Phan, J. Mathon, D.M. Edwards and R.B. Muniz, J. Magn. Magn. Mater. 104-107 (1992) 1876. [2] R. Swirkowicz, Physica B 203 (1994) 9. [3] D. Mercier and J.C.S. Levy, M.L. Watson, J.S.S. Whiting and A. Chambers, Phys. Rev. B 43 (1991) 3311. [4] A.Z. Maksymowicz, K. Zakrzewska, J.S.S. Whiting, M.L. Watson, A. Chambers and S. Kaprzyk, J. Magn. Magn. Mater. 127 (1993) 219. [5] H. Nakanishi, T. Tamura, H. Kasai and A. Okiji, J. Magn. Magn. Mater. 126 (1993) 394.
•1•
~i ELSEVIER
Journalof magnetism and magnetic materials
Spin waves in thin films composed of ferromagnetic and nonmagnetic materials R. S w i r k o w i c z
*
Institute of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland Abstract Systems composed of Ni and nonmagnetic layers are studied within the framework of the multiband approach. Spin-wave spectra and the amplitudes of modes are investigated. Changes in the strength of the effective coupling between Ni layers are discussed.
I. Introduction
Recently, films composed of magnetic and nonmagnetic materials have been studied. Changes of thickness of a nonmagnetic spacer lead to changes in the strength of the magnetic coupling and substantially influence the spectrum of collective modes. The spin-wave formalism within the framework of the band model was used to determine the coupling between ferromagnetic monolayers embedded in nonmagnetic metal [1]. Problems of the dynamic susceptibility and spin waves in double layers were investigated [2]. In the present paper, structures composed of nickel and nonmagnetic transition metals are studied. 2. The model
Systems composed of Ni sublayers which are separated by k nonmagnetic planes are investigated. Calculations are performed within the framework of the multiband model with five d-bands taken into account. The tight-binding approximation with Slater-Koster parameterization is used. Parameters appropriate for Ni and nonmagnetic metals (Cu, Au, Ag) are taken. The one-electron spin-independent potential is chosen in such a way that the correct number of d-electrons in both metals is obtained. The Coulomb integral U equal to 1.0 eV is chosen in all layers. The surface of the film and the interfaces are perpendicular to the [001] direction. The band structure is calculated in the Hartree-Fock approximation and profiles of the ground-state magnetization are found. The magnetization is slightly enhanced in
Ni layers corresponding to the surfaces, but it is lowered at Ni-nonmagnetic metal interfaces. A very small magnetic moment is induced in the planes of nonmagnetic metals which are in direct contact with Ni. Spin waves are investigated within the framework of the Random Phase Approximation using the dynamic susceptibility method. The transverse susceptibility is calculated in the mixed Bloch-Wannier representation [1,2] for various values of a two-dimensional wave vector q parallel to the interfaces. 3. Spin waves. Results and discussion
Spin waves are investigated for systems which are composed of two Ni sublayers, each containing N = 5 atomic planes, and separated by k sheets of non-magnetic metal. The thickness of the spacer is varied from k = 1 to 7 atomic layers. In the spin-wave spectrum, pairs of modes with close energies are found. The differences in energy corresponding to two pairs of low-lying modes are depicted in Fig. 1 as a function of k. Calculations are performed for a Cu spacer and q = 0. The energy gap between two branches is a distinctive feature of a given pair and it diminishes with an increase in k. For k = 1 the
m
t'!
w
21D
k
* Fax: 48-2-628-21-71;email: [email protected].
o
"t'4"4" k
Fig. 1. Energy gaps for the first (a) and the second (b) pairs of low-lying modes in the case of a double layer versus k (q = 0).
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0304-8853(94)01561-9
1946
R. Swirkowicz /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1945-1946
~ ~ ~ ( 1 . 0 )
Fig. 2. Dispersion relations for a double layer with Ag spacer. gap corresponding to the second pair is considerably greater than for the first. The energy gap between the two lowest modes with q = 0 can be considered as a measure of the indirect exchange coupling between ferromagnetic films [1]. Therefore, in the systems under consideration the effective exchange coupling strongly diminishes when the thickness of the spacer increases from 1 to 7 atomic layers. Spin-wave dispersion relations are calculated for systems composed of two Ni sublayers separated by k = 3 spacer planes. The parameters are modified in such a way that the spacer corresponding to Cu, Au or Ag is chosen. The general course of the dispersion curves is similar in all cases. In the region of small q, pairs of modes with close energies are found (Fig. 2). The distance in energy for a given pair is greater for higher q. The amplitudes of a pair of modes are practically the same. In Ni planes which are in direct contact with the nonmagnetic spacer the amplitudes are relatively high, whereas in the spacer layers they are negligibly small. A considerable discontinuity appears near the boundary, which is well seen for antisymmetric modes. Similar calculations are performed for systems composed of 3 Ni sublayers. Dispersion curves obtained for low-lying modes are presented in Fig. 3 for the case of a Cu spacer and k = 1. The subband which consists of three
modes is well seen in the figure. Two of the modes in the group are mainly localized in the outer Ni sublayers, whereas the third one is essentially confined in the central part of the film. Because of damping effects, it is difficult to determine precisely the energies of high-lying modes. Only some points corresponding to the higher subbands are presented in the figure. Energies characteristic for spin waves in a separate Ni film consisting of N = 5 atomic planes are marked in the figure by bars. There is a consistency between values of energies in both systems. Therefore, the modes have a tendency to form groups consisting of several branches with energy values close to the ones characteristic for a separate film. Such results show that the effective magnetic interactions between Ni sublayers are weak. Next, calculations are performed for a system composed of 5 Ni sublayers separated by Cu spacers with k = 2. In the region of small q, the five lowest modes are practically degenerated. Two of the modes are localized in Ni sublayers close to the surfaces. Three other modes in the group have considerable amplitudes in inner Ni sublayers. In the spin-wave spectrum two subbands can be seen. Results presented in this paper are consistent with those obtained in Heisenberg and phenomenological approaches [3,4]. Pairs of modes with close energies appear to be a characteristic feature of the spin-wave spectrum of double films and are observed experimentally [3,4]. Relatively high amplitudes and a considerable discontinuity of antisymmetric modes at the interfaces are obtained in the Heisenberg model [3]. However, compared to the results of the phenomenological approach [4] it seems that the calculated effective coupling between magnetic layers diminishes too fast in the case of thicker interlayers. It is possible that in the systems under consideration, electrons of s symmetry play an important role and influence the effective coupling between magnetic sublayers for the thick spacer [5]. Acknowledgement: The work was supported by the Scientific Research Committee, project No 2 2321 9102.
References
10oi ,o)
Fig. 3. Dispersion relations for three Ni sublayers with Cu spacer.
[1] M.S. Phan, J. Mathon, D.M. Edwards and R.B. Muniz, J. Magn. Magn. Mater. 104-107 (1992) 1876. [2] R. Swirkowicz, Physica B 203 (1994) 9. [3] D. Mercier and J.C.S. Levy, M.L. Watson, J.S.S. Whiting and A. Chambers, Phys. Rev. B 43 (1991) 3311. [4] A.Z. Maksymowicz, K. Zakrzewska, J.S.S. Whiting, M.L. Watson, A. Chambers and S. Kaprzyk, J. Magn. Magn. Mater. 127 (1993) 219. [5] H. Nakanishi, T. Tamura, H. Kasai and A. Okiji, J. Magn. Magn. Mater. 126 (1993) 394.