- Email: [email protected]
Quantum wells in trilayers: Dependence of the properties on the thickness of magnetic and nonmagnetic layers
Journal of Magnetism and Magnetic Materials 156 (1996) 193-194
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Quantum wells in trilayers: dependence of the properties on the thickness of magnetic and nonmagnetic layers V . M . U z d i n a N . S . Y a r t s e v a b,* a Institute of Chemistry, St-Petersburg University, St Petersburg 198904, Russia b Institute of Metal Physics, ul. S. Kot~aleuskoy 18, GSP-170, Ekaterinburg 620219, Russia
Abstract Electronic systems of magnetic multilayers have been considered within the framework of the simplest quantum well models. The exchange coupling energy for trilayers has been calculated. It is shown that with varying magnetic layer thickness, the exchange coupling strength changes in a different way than in the case of nonmagnetic spacer thickness variation. This is explained in terms of the difference in the number of levels in quantum wells for ferromagnetic and antiferromagnetic structures.
The oscillatory behavior of the exchange coupling between ferromagnetic layers separated by a nonmagnetic spacer is broadly discussed. Recent experiments [1,2] have confirmed that the oscillations may not only be a function of thickness of the spacer layer, but also of those of the magnetic layers. The nature of the phenomenon can be clarified within the framework of the model presupposing the confinement of the itinerant electrons in the quantum wells tbrmed by the magnetic films [3-5]. The structure of the wells is different for electrons of opposite spin. The oscillation of exchange coupling is connected with the change in the number of quantum states for ferromagnetic (FM) and antiferromagnetic (AFM) ordering in the superlattice, depending on the thicknesses of the magnetic and nonmagnetic layers. Changing the density of states at the Fermi level leads to giant magnetoresistance. We consider a trilayer consisting of two magnetic layers with thicknesses Lj and L 2, separated by a nonmagnetic spacer of thickness L o. For simplicity, we assume that all states of one spin orientation of the magnetic layers are occupied. Itinerant electrons with the same spin orientation in the spacer cannot transfer easily through the magnetic layers, and appear to be in quantum wells. For FM ordering of magnetic layers, the electrons with one spin orientation are in a quantum well of width L 0, and L o + L l + L 2 for the other spin. For AFM ordering the quantum well widths are L o + L~ and L o + L 2, respectively (see Fig. 1).
" Corresponding author. [email protected].
Fax:
+7-3432-445244;
e-mail:
In the layer plane the electrons move freely with energy h '")k q2l / 2 m , where kbl is the in-plane wavevector and m is the electron mass. For the perpendicular direction the electrons are confined to the potential well. In the simplest case of an unlimited square-well potential the energy of the electrons per unit interface area per well can be expressed as follows [5]:
×
30
]'
where L is a width of a well, n L is the number of transverse quantum levels below the Fermi level e v. We assume that the Fermi energy of the system is fixed by a large number of d-electrons in the magnetic layers. In this case the total number of itinerant electrons N is not fixed and is different for FM and AFM ordering. To compare the energies of the FM and AFM states we have to consider the thermodynamic potentials O = E ~ . - e v N. For a single well we have [5]
Fig. 2 shows the difference A,(2 between the potential of the electrons in the quantum well and that for the electrons without energy quantization, as a function of L. A ~ oscillates with L and these oscillations have non-sinusoidal
0304-8853/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 8 8 5 3 ( 9 5 ) 0 0 7 5 4 - 7
V.M. Uzdin, N.S. Yartseva/Journal of Magnetism and Magnetic Materials 156 (1996) 193-194
194
15
Lo + L1 + L2
£
L2
Fig. 1. Quantum wells for FM and AFM ordering in a trilayer.
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/~
L(A)
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character: the increase in the function during the period is faster than the decrease. It is easy to see that enhancement of A [2 takes place near the widths where the new quantum level arises. The magnitude of the oscillations decreases markedly with L. This means that the smaller the width of the well in the multilayer, the greater is its contribution to the exchange coupling energy. Let us now consider the exchange coupling in the trilayer. The difference between the energy in FM and AFM states is a function not only of L o but also of L l and L 2. However, with varying magnetic layer thickness, the exchange coupling strength changes in a different way than in the case when the nonmagnetic spacer thickness is varied. This can be explained in terms of the difference A(L o, Ll, L 2) in the number of levels in the quantum wells for the FM and AFM structures. The change in the nonmagnetic spacer thickness results in an alterations of the width of the quantum wells in both the FM and AFM states. In the symmetrical case, the new quantum levels arise in both wells, so that the number of quantum levels changes by two. The range of variation of L o splits into two regions: over the first region, A = + 1, whereas over the second region A = - 1. In the asymmetric case the region with A = 0 can lie between the regions mentioned above. The main contribution to the exchange coupling is given by the narrowest well, i.e. the well with width L o for FM ordering. Thus, changing L o leads to oscillations of the coupling analogous to those in A g2. Such behaviour is scarcely affected by changing the width of other wells. The exchange coupling oscillates near zero, and its maximum and minimum correlate with positive and negative A, respectively.
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t"--~-~,,
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10
11
12
13
14.
15
16
17
18
19
Lt
Fig. 3. Dependence of ~+~AFM -- aC~FM (meV/a.u.) on L0 (,~) and L I (A) for M (32 A) NM (L o) M (L I) trilayer (~v = 10 eV).
If the thickness of a magnetic layer (L]) is altered, the width of only one quantum well changes in both the FM and the AFM states; A is successively equal to either 0 and + 1, or 0 and - 1, depending on the thickness of the unchanged layers. The contribution to the exchange coupling of the electrons in the narrowest well now does not change. It can be positive or negative, depending on L 0. In the first case the exchange coupling oscillates near its positive average value, while in the second the oscillations take place near the negative value. The magnitude of the oscillations is much lower than if the width of the nonmagnetic spacer changes. The electrons of the well L o + Lj for the AFM state now play the main role in oscillations of the exchange coupling. If the amplitude of oscillations near L = L l + L o in Fig. 2 is less than the value A J2 for L = L o, there are the oscillations of the exchange coupling without a change of sign. The results of the calculations of the exchange coupling strength versus L o and L l are shown in Fig. 3, where the lines correspond to the constant / 2 A ~ --/2~M. Oscillations of the exchange coupling have been obtained in recent experiments with asymmetric Co(32 A)/Ru(LRu)/Co(Lco) trilayers by means of ferromagnetic resonance [2]. The results of these measurements are in qualitative agreement with our theory. Acknowledgements: The research described in this publication was made possible in part by Grants NR 62300 and NMK000 from the ISF and the Russian Government. References
-2
- 1 0 I-
Fig. 2. A/2 (meV/a.u.) and number of quantum levels in the well versus L.
[1] S.N. Okuno and K. Inomata, Phys. Rev. Lett. 72 (1994) 1553. [2] Z. Zhang, L. Zhou, P.E. Wigen and K. Ounadjela, Phys. Rev. B 50 (1994) 6094. [3] J. Barnas, J. Magn. Magn. Mater. 111 (1992) L215. [4] P. Bruno, Europhys. Lett. 23 (1993) 615, [5] A.K. Kazansky and V.M. Uzdin, J. Magn. Magn. Mater. 138 (1994) 287.
"'"
,-o,,nellsm
Journal of
nlun011¢ , ~
ELSEVIER
Ilmleflall
Quantum wells in trilayers: dependence of the properties on the thickness of magnetic and nonmagnetic layers V . M . U z d i n a N . S . Y a r t s e v a b,* a Institute of Chemistry, St-Petersburg University, St Petersburg 198904, Russia b Institute of Metal Physics, ul. S. Kot~aleuskoy 18, GSP-170, Ekaterinburg 620219, Russia
Abstract Electronic systems of magnetic multilayers have been considered within the framework of the simplest quantum well models. The exchange coupling energy for trilayers has been calculated. It is shown that with varying magnetic layer thickness, the exchange coupling strength changes in a different way than in the case of nonmagnetic spacer thickness variation. This is explained in terms of the difference in the number of levels in quantum wells for ferromagnetic and antiferromagnetic structures.
The oscillatory behavior of the exchange coupling between ferromagnetic layers separated by a nonmagnetic spacer is broadly discussed. Recent experiments [1,2] have confirmed that the oscillations may not only be a function of thickness of the spacer layer, but also of those of the magnetic layers. The nature of the phenomenon can be clarified within the framework of the model presupposing the confinement of the itinerant electrons in the quantum wells tbrmed by the magnetic films [3-5]. The structure of the wells is different for electrons of opposite spin. The oscillation of exchange coupling is connected with the change in the number of quantum states for ferromagnetic (FM) and antiferromagnetic (AFM) ordering in the superlattice, depending on the thicknesses of the magnetic and nonmagnetic layers. Changing the density of states at the Fermi level leads to giant magnetoresistance. We consider a trilayer consisting of two magnetic layers with thicknesses Lj and L 2, separated by a nonmagnetic spacer of thickness L o. For simplicity, we assume that all states of one spin orientation of the magnetic layers are occupied. Itinerant electrons with the same spin orientation in the spacer cannot transfer easily through the magnetic layers, and appear to be in quantum wells. For FM ordering of magnetic layers, the electrons with one spin orientation are in a quantum well of width L 0, and L o + L l + L 2 for the other spin. For AFM ordering the quantum well widths are L o + L~ and L o + L 2, respectively (see Fig. 1).
" Corresponding author. [email protected].
Fax:
+7-3432-445244;
e-mail:
In the layer plane the electrons move freely with energy h '")k q2l / 2 m , where kbl is the in-plane wavevector and m is the electron mass. For the perpendicular direction the electrons are confined to the potential well. In the simplest case of an unlimited square-well potential the energy of the electrons per unit interface area per well can be expressed as follows [5]:
×
30
]'
where L is a width of a well, n L is the number of transverse quantum levels below the Fermi level e v. We assume that the Fermi energy of the system is fixed by a large number of d-electrons in the magnetic layers. In this case the total number of itinerant electrons N is not fixed and is different for FM and AFM ordering. To compare the energies of the FM and AFM states we have to consider the thermodynamic potentials O = E ~ . - e v N. For a single well we have [5]
Fig. 2 shows the difference A,(2 between the potential of the electrons in the quantum well and that for the electrons without energy quantization, as a function of L. A ~ oscillates with L and these oscillations have non-sinusoidal
0304-8853/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 8 8 5 3 ( 9 5 ) 0 0 7 5 4 - 7
V.M. Uzdin, N.S. Yartseva/Journal of Magnetism and Magnetic Materials 156 (1996) 193-194
194
15
Lo + L1 + L2
£
L2
Fig. 1. Quantum wells for FM and AFM ordering in a trilayer.
L\o.~J
7 6 ~
/~
L(A)
>
W
\o>
~o.~'
~3
o.__._~
12
. J ~
I(. ""-,
character: the increase in the function during the period is faster than the decrease. It is easy to see that enhancement of A [2 takes place near the widths where the new quantum level arises. The magnitude of the oscillations decreases markedly with L. This means that the smaller the width of the well in the multilayer, the greater is its contribution to the exchange coupling energy. Let us now consider the exchange coupling in the trilayer. The difference between the energy in FM and AFM states is a function not only of L o but also of L l and L 2. However, with varying magnetic layer thickness, the exchange coupling strength changes in a different way than in the case when the nonmagnetic spacer thickness is varied. This can be explained in terms of the difference A(L o, Ll, L 2) in the number of levels in the quantum wells for the FM and AFM structures. The change in the nonmagnetic spacer thickness results in an alterations of the width of the quantum wells in both the FM and AFM states. In the symmetrical case, the new quantum levels arise in both wells, so that the number of quantum levels changes by two. The range of variation of L o splits into two regions: over the first region, A = + 1, whereas over the second region A = - 1. In the asymmetric case the region with A = 0 can lie between the regions mentioned above. The main contribution to the exchange coupling is given by the narrowest well, i.e. the well with width L o for FM ordering. Thus, changing L o leads to oscillations of the coupling analogous to those in A g2. Such behaviour is scarcely affected by changing the width of other wells. The exchange coupling oscillates near zero, and its maximum and minimum correlate with positive and negative A, respectively.
\o>'
U '~'-,
C"-"~,,.
t"--~-~,,
t°'~-'-. O.3
10
11
12
13
14.
15
16
17
18
19
Lt
Fig. 3. Dependence of ~+~AFM -- aC~FM (meV/a.u.) on L0 (,~) and L I (A) for M (32 A) NM (L o) M (L I) trilayer (~v = 10 eV).
If the thickness of a magnetic layer (L]) is altered, the width of only one quantum well changes in both the FM and the AFM states; A is successively equal to either 0 and + 1, or 0 and - 1, depending on the thickness of the unchanged layers. The contribution to the exchange coupling of the electrons in the narrowest well now does not change. It can be positive or negative, depending on L 0. In the first case the exchange coupling oscillates near its positive average value, while in the second the oscillations take place near the negative value. The magnitude of the oscillations is much lower than if the width of the nonmagnetic spacer changes. The electrons of the well L o + Lj for the AFM state now play the main role in oscillations of the exchange coupling. If the amplitude of oscillations near L = L l + L o in Fig. 2 is less than the value A J2 for L = L o, there are the oscillations of the exchange coupling without a change of sign. The results of the calculations of the exchange coupling strength versus L o and L l are shown in Fig. 3, where the lines correspond to the constant / 2 A ~ --/2~M. Oscillations of the exchange coupling have been obtained in recent experiments with asymmetric Co(32 A)/Ru(LRu)/Co(Lco) trilayers by means of ferromagnetic resonance [2]. The results of these measurements are in qualitative agreement with our theory. Acknowledgements: The research described in this publication was made possible in part by Grants NR 62300 and NMK000 from the ISF and the Russian Government. References
-2
- 1 0 I-
Fig. 2. A/2 (meV/a.u.) and number of quantum levels in the well versus L.
[1] S.N. Okuno and K. Inomata, Phys. Rev. Lett. 72 (1994) 1553. [2] Z. Zhang, L. Zhou, P.E. Wigen and K. Ounadjela, Phys. Rev. B 50 (1994) 6094. [3] J. Barnas, J. Magn. Magn. Mater. 111 (1992) L215. [4] P. Bruno, Europhys. Lett. 23 (1993) 615, [5] A.K. Kazansky and V.M. Uzdin, J. Magn. Magn. Mater. 138 (1994) 287.