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Influence of the magnetic slab thickness on the exchange coupling in a trilayer
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Influenceof the magneticslabthicknesson t couplingin a tdayer S. Krompiewski Institute of Mokcdar
*
Physics, Polish Academy of Sciences, Pow&, Poland
AbStFaCt Exchange coupling between two magnetic slabs separated by a nonmagnetic spacer is shown to depeud magnetic slab thickness. Systematic studies of trilayers AII,/BN/AM (where M and N stand for the num and nonmagnetic monoIayers, respectively), have been carried out within the single baud cubed (IV, Ml-plane a ‘phase diagram’ has been plotted to visualize multiple chauges in the exchange coupling sign.
Quite recently more and more asention has been devoted to the effect of the ferromagnetic slab thickness on oscillatory behaviour of exchange coupling, J, in mul!ilayers. In contrast to the spacer thickness-dependence of the coupling which has been well understood now in terms of the RKKY-type theory 111, tight-biidiig model 121, first principles calculations [3-5] and some other approaches, the magnetic slab thickness-dependence of J is still under diiiiie. ihe relevance of the magnetic slab thickness was first emphasized in Refs. [6] and [4] independently, in terms of different theoretical approaches. Recently a transparent theory of this phenomenon was formulated in Ref. [7] within the RKKY+type theory, and the present status of the problem was presented in Ref. IS]. On one hand, it is now well-known that the magnetic layer band structure determines the oscillatory behaviour of the exchange coupling versus the spacer thickness [S]. On the other hand it seems that there is no more doubt that the magnetic slab thickness itself is quite crucial and influences strongly the exchange coupling. This conclusion is theoretical motielindependent and it has been already proved experimentally [9,10]. Both the theoreticalas well as experimental evidences show clearly that J does oscillate even if the thickness of a ferromaguet gets changed with the spacer thickness kept fixed. The aim of the present paper is to carry out systematic studies of trilayers AJBJA,,, composed of hvo ferromagnetic slabs A, separated by a nonmagnetic spacer 5. There are M monolayers belonging to A aud N monalayers belonging to B. The question to answer is how do M
* Fax: Pl.
+48-61-674751;
J can be discussed in terms of the sizequotation effect. if the electron is confined to L atomic plaues in the zdirecticn, its energy and eigen-function are: q,,
0304-8853/9S/$O!UO 8 1995 Ekvier SSDIO304-8853(94)01541-4
7=n?T/(L+l), = c SiQ(r.
U,(T)
email: [email protected].
Science B.V. Au rights reserved
$9 7) =
-q
cos k,+cos
l
k,+coss),
(1) 12)
1ofMagnetism ddniagnetic
. . .. .. . . . . : : : .. : : : : : . .
. . .
: . . .
:.
: : .
.t
: : . .
. . .
. .
. .
. .
l . l
. .
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: . .
: . .
: . .
:.
. .
: . .
: . .
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: l .
: l.
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. : : : .. . : .: . :.. 2. .: . .* . . . . . . . . :. .:. 1
:
:.
:.
Fig. 1. Sign of the exchange ccmp!ing of the A, /BN/AM sign is negative for Ad, N marked with a dot and E, = -3 (a‘a, E, = - 1.9 (b). (see e.g. Ref. Ill] for details), and additionally local merits per lattice site k&ding those induced ) have been found from m, = nrr - n, 1 with
(3) E, is the Fermi energy, 8 is the step function and tegration over the Zdimensional Briltouin zone (BZ) rmed by means of the mean-value point 1123 with more than 4 X 10’ k-points in the quarter ts of the present studies are presented in Figs. shows mu~~~~e crossovers from the ft to ens kmd vice versa) on the (N, Ml-plane. race of tk exchange coupling is seen to be to the magnetic layer thickness M. This th to Fig. Ia as well as to Fig. lb, although in
Fig 2. ~xsb~ge fxwpairtg of the tdayers versns M for: E, - - 3 M = 7, 8,9 (a); and E, = - 1.9 with N = 8, 13 (b).
Materials 140-W
~1995) 515-516
the latter case the N-dependence of J starts to be only slightly M-dependent for roughly M > 15. The periods of oscillations with N are 3 and roughly 10 for iE, = -3 and - 1.9, respectively, and are connected with extremal cross-sections of the corresponding Fermi surfaces [2]. If N is fixed and M is allowed to vary, the exchange can always remain either positive or negative or changes its sign. It is quite crucial Fc, know the regions where the coupling is negative since one can expect a giant magnetoresistance there. In order to see in more detail how J behaves for fixed N, Fig. 2 has been plotted. Interestingly enough the periods of oscillations versus M are equal to those versus N. The curves shown in Fig. 2 resemble results obtained in [7] by a diffeaent method, what proves that the phenomenon is universal and model-independent. The coincidence of the oscillation periods versus N and M is however incidental and should not be expected if a lattice constant of the ferromagnet was different from that of the spacer (cf. Ref. [5]). As regards magnetic moments, it has been found that ferromagnetic slabs have enhanced moments at the interface by roughly 10% while the spacer gets induced magnetic moments relatively big at the interface and vanishing towards the spacer centre. In general it is known from ab initio computations that the interface ferromagnetic layers may have both enhanced magnetization [4] as well as reduced one ES], depending cn the metals a multilayer consists of. In conclusion, it has been shown that the ferromagnetic layer thickness has a considerable effect on the oscillatory behavour of the exchange coupling. The oscillations versus M are essentially as pronounced as those versus N and are also due to the size quantization effect. References [l] P. Bruno and C. Chappert, Phys. Rev. B 46 (1991) 261. [2] D.M. Edwards, J. Mathon, R.B. Muniz and MS. Phan, J. Phys. C: Cond. Matt. 3 (1991) 4941. [3] F. Herman, J. Sticht and M. Van Schilfgaarde, J. Appl. Phys. 69 (1991) 4783. [4] S. Krompiewski, U. Krey and J. Pimay, J. Magn. Magn. Mater. 121 (1993) 238. [5] S. Krompiewski, F. Swiss and U. Krey, E. Phys. Lett. 26 (1994) 303. [6] J. Barnag, J. Magn. Magn. Mater. 111 (1992) L215. [7] P. Bruno, E. Phys. Lett. 23 (1993) 615. [8] P. Lang, L. Nordsttim, R. Zeller and P.H. Dederichs, Phys. Rev. L&t. 71 (1993) 1927. [9] P.J.H. Bloemen, M.T. Johnson, M.T.H. van de Vorst, R. Coehoorn. J.J. de Vries R. Jungblut, J. aan de Stegge, A. Reinders and W.J.M de Jonge, Phys. Rev. Lctt. 72 (1994) [lQ] ii: Okuno and K. InomaFa, Phys. Rev. L&t. 72 (1994) 1553. Ill] J. Barn&, J. Magn. Magn. Mater. 128 (1993) 171. [12] S.L. Cunningham, Phys. Rev. B 10 (1974) 4988.
Influenceof the magneticslabthicknesson t couplingin a tdayer S. Krompiewski Institute of Mokcdar
*
Physics, Polish Academy of Sciences, Pow&, Poland
AbStFaCt Exchange coupling between two magnetic slabs separated by a nonmagnetic spacer is shown to depeud magnetic slab thickness. Systematic studies of trilayers AII,/BN/AM (where M and N stand for the num and nonmagnetic monoIayers, respectively), have been carried out within the single baud cubed (IV, Ml-plane a ‘phase diagram’ has been plotted to visualize multiple chauges in the exchange coupling sign.
Quite recently more and more asention has been devoted to the effect of the ferromagnetic slab thickness on oscillatory behaviour of exchange coupling, J, in mul!ilayers. In contrast to the spacer thickness-dependence of the coupling which has been well understood now in terms of the RKKY-type theory 111, tight-biidiig model 121, first principles calculations [3-5] and some other approaches, the magnetic slab thickness-dependence of J is still under diiiiie. ihe relevance of the magnetic slab thickness was first emphasized in Refs. [6] and [4] independently, in terms of different theoretical approaches. Recently a transparent theory of this phenomenon was formulated in Ref. [7] within the RKKY+type theory, and the present status of the problem was presented in Ref. IS]. On one hand, it is now well-known that the magnetic layer band structure determines the oscillatory behaviour of the exchange coupling versus the spacer thickness [S]. On the other hand it seems that there is no more doubt that the magnetic slab thickness itself is quite crucial and influences strongly the exchange coupling. This conclusion is theoretical motielindependent and it has been already proved experimentally [9,10]. Both the theoreticalas well as experimental evidences show clearly that J does oscillate even if the thickness of a ferromaguet gets changed with the spacer thickness kept fixed. The aim of the present paper is to carry out systematic studies of trilayers AJBJA,,, composed of hvo ferromagnetic slabs A, separated by a nonmagnetic spacer 5. There are M monolayers belonging to A aud N monalayers belonging to B. The question to answer is how do M
* Fax: Pl.
+48-61-674751;
J can be discussed in terms of the sizequotation effect. if the electron is confined to L atomic plaues in the zdirecticn, its energy and eigen-function are: q,,
0304-8853/9S/$O!UO 8 1995 Ekvier SSDIO304-8853(94)01541-4
7=n?T/(L+l), = c SiQ(r.
U,(T)
email: [email protected].
Science B.V. Au rights reserved
$9 7) =
-q
cos k,+cos
l
k,+coss),
(1) 12)
1ofMagnetism ddniagnetic
. . .. .. . . . . : : : .. : : : : : . .
. . .
: . . .
:.
: : .
.t
: : . .
. . .
. .
. .
. .
l . l
. .
:.
: . .
: . .
: . .
:.
. .
: . .
: . .
: . .
: . .
: l .
: l.
: . .
:
. : : : .. . : .: . :.. 2. .: . .* . . . . . . . . :. .:. 1
:
:.
:.
Fig. 1. Sign of the exchange ccmp!ing of the A, /BN/AM sign is negative for Ad, N marked with a dot and E, = -3 (a‘a, E, = - 1.9 (b). (see e.g. Ref. Ill] for details), and additionally local merits per lattice site k&ding those induced ) have been found from m, = nrr - n, 1 with
(3) E, is the Fermi energy, 8 is the step function and tegration over the Zdimensional Briltouin zone (BZ) rmed by means of the mean-value point 1123 with more than 4 X 10’ k-points in the quarter ts of the present studies are presented in Figs. shows mu~~~~e crossovers from the ft to ens kmd vice versa) on the (N, Ml-plane. race of tk exchange coupling is seen to be to the magnetic layer thickness M. This th to Fig. Ia as well as to Fig. lb, although in
Fig 2. ~xsb~ge fxwpairtg of the tdayers versns M for: E, - - 3 M = 7, 8,9 (a); and E, = - 1.9 with N = 8, 13 (b).
Materials 140-W
~1995) 515-516
the latter case the N-dependence of J starts to be only slightly M-dependent for roughly M > 15. The periods of oscillations with N are 3 and roughly 10 for iE, = -3 and - 1.9, respectively, and are connected with extremal cross-sections of the corresponding Fermi surfaces [2]. If N is fixed and M is allowed to vary, the exchange can always remain either positive or negative or changes its sign. It is quite crucial Fc, know the regions where the coupling is negative since one can expect a giant magnetoresistance there. In order to see in more detail how J behaves for fixed N, Fig. 2 has been plotted. Interestingly enough the periods of oscillations versus M are equal to those versus N. The curves shown in Fig. 2 resemble results obtained in [7] by a diffeaent method, what proves that the phenomenon is universal and model-independent. The coincidence of the oscillation periods versus N and M is however incidental and should not be expected if a lattice constant of the ferromagnet was different from that of the spacer (cf. Ref. [5]). As regards magnetic moments, it has been found that ferromagnetic slabs have enhanced moments at the interface by roughly 10% while the spacer gets induced magnetic moments relatively big at the interface and vanishing towards the spacer centre. In general it is known from ab initio computations that the interface ferromagnetic layers may have both enhanced magnetization [4] as well as reduced one ES], depending cn the metals a multilayer consists of. In conclusion, it has been shown that the ferromagnetic layer thickness has a considerable effect on the oscillatory behavour of the exchange coupling. The oscillations versus M are essentially as pronounced as those versus N and are also due to the size quantization effect. References [l] P. Bruno and C. Chappert, Phys. Rev. B 46 (1991) 261. [2] D.M. Edwards, J. Mathon, R.B. Muniz and MS. Phan, J. Phys. C: Cond. Matt. 3 (1991) 4941. [3] F. Herman, J. Sticht and M. Van Schilfgaarde, J. Appl. Phys. 69 (1991) 4783. [4] S. Krompiewski, U. Krey and J. Pimay, J. Magn. Magn. Mater. 121 (1993) 238. [5] S. Krompiewski, F. Swiss and U. Krey, E. Phys. Lett. 26 (1994) 303. [6] J. Barnag, J. Magn. Magn. Mater. 111 (1992) L215. [7] P. Bruno, E. Phys. Lett. 23 (1993) 615. [8] P. Lang, L. Nordsttim, R. Zeller and P.H. Dederichs, Phys. Rev. L&t. 71 (1993) 1927. [9] P.J.H. Bloemen, M.T. Johnson, M.T.H. van de Vorst, R. Coehoorn. J.J. de Vries R. Jungblut, J. aan de Stegge, A. Reinders and W.J.M de Jonge, Phys. Rev. Lctt. 72 (1994) [lQ] ii: Okuno and K. InomaFa, Phys. Rev. L&t. 72 (1994) 1553. Ill] J. Barn&, J. Magn. Magn. Mater. 128 (1993) 171. [12] S.L. Cunningham, Phys. Rev. B 10 (1974) 4988.