Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers

The exchange coupling of magnetically ordered layers (MOLs) through a nonmagnetic metallic spacer was calculated. The induced magnetization in the spacer, taking into account the influence of an external magnetic field, was calculated, too. This calculation shows that the energy of coupling of th...

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Дата:	2004
Автор:	Gorobets, V.Yu.
Формат:	Стаття
Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2004
Назва видання:	Физика низких температур
Теми:	Низкотемпеpатуpный магнетизм
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/120046
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Цитувати:	Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers / V.Yu. Gorobets // Физика низких температур. — 2004. — Т. 30, № 10. — С.1045–1052. — Бібліогр.: 31 назв. — англ.

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spelling	irk-123456789-1200462017-06-11T03:05:09Z Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers Gorobets, V.Yu. Низкотемпеpатуpный магнетизм The exchange coupling of magnetically ordered layers (MOLs) through a nonmagnetic metallic spacer was calculated. The induced magnetization in the spacer, taking into account the influence of an external magnetic field, was calculated, too. This calculation shows that the energy of coupling of the MOLs through the nonmagnetic metallic spacer is a long-periodic function of the spacer’s thickness and magnetic field, i.e., the exchange coupling between the layers varies from ferromagnetic to antiferromagnetic and vice versa depending on the spacer’s thickness and magnetic field. Also this calculation shows that in nonferromagnetic spacer the induced magnetization can undergo many complete rotations depending on distance to the boundaries with the MOLs. Moreover, absolute value of induced magnetization nonmonotonously decays with distance from the interfaces inside the spacer. It is shown that the character of the decay of absolute value magnetization from the interfaces into the interior of the spacer is influenced by magnetic field. 2004 Article Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers / V.Yu. Gorobets // Физика низких температур. — 2004. — Т. 30, № 10. — С.1045–1052. — Бібліогр.: 31 назв. — англ. 0132-6414 PACS: 75.70.–i http://dspace.nbuv.gov.ua/handle/123456789/120046 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм
spellingShingle	Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм Gorobets, V.Yu. Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers Физика низких температур
description	The exchange coupling of magnetically ordered layers (MOLs) through a nonmagnetic metallic spacer was calculated. The induced magnetization in the spacer, taking into account the influence of an external magnetic field, was calculated, too. This calculation shows that the energy of coupling of the MOLs through the nonmagnetic metallic spacer is a long-periodic function of the spacer’s thickness and magnetic field, i.e., the exchange coupling between the layers varies from ferromagnetic to antiferromagnetic and vice versa depending on the spacer’s thickness and magnetic field. Also this calculation shows that in nonferromagnetic spacer the induced magnetization can undergo many complete rotations depending on distance to the boundaries with the MOLs. Moreover, absolute value of induced magnetization nonmonotonously decays with distance from the interfaces inside the spacer. It is shown that the character of the decay of absolute value magnetization from the interfaces into the interior of the spacer is influenced by magnetic field.
format	Article
author	Gorobets, V.Yu.
author_facet	Gorobets, V.Yu.
author_sort	Gorobets, V.Yu.
title	Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers
title_short	Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers
title_full	Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers
title_fullStr	Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers
title_full_unstemmed	Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers
title_sort	magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2004
topic_facet	Низкотемпеpатуpный магнетизм
url	http://dspace.nbuv.gov.ua/handle/123456789/120046
citation_txt	Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers / V.Yu. Gorobets // Физика низких температур. — 2004. — Т. 30, № 10. — С.1045–1052. — Бібліогр.: 31 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT gorobetsvyu magnetizationofanonferromagneticmetalspacersandwichedbetweentwomagneticallyorderedlayers
first_indexed	2025-07-08T17:08:41Z
last_indexed	2025-07-08T17:08:41Z
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fulltext	Fizika Nizkikh Temperatur, 2004, v. 30, No. 10, p. 1045–1052 Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers V.Yu. Gorobets Institute of Magnetism of the National Academy of Sciences of Ukraine 36-b Vernadsky Ave. Kiev 03142, Ukraine E-mail: gorob@mail.kar.net Received January 6, 2004, revised April 1, 2004 The exchange coupling of magnetically ordered layers (MOLs) through a nonmagnetic metallic spacer was calculated. The induced magnetization in the spacer, taking into account the influence of an external magnetic field, was calculated, too. This calculation shows that the energy of cou- pling of the MOLs through the nonmagnetic metallic spacer is a long-periodic function of the spacer’s thickness and magnetic field, i.e., the exchange coupling between the layers varies from ferromagnetic to antiferromagnetic and vice versa depending on the spacer’s thickness and mag- netic field. Also this calculation shows that in nonferromagnetic spacer the induced magnetization can undergo many complete rotations depending on distance to the boundaries with the MOLs. Moreover, absolute value of induced magnetization nonmonotonously decays with distance from the interfaces inside the spacer. It is shown that the character of the decay of absolute value mag- netization from the interfaces into the interior of the spacer is influenced by magnetic field. PACS: 75.70.–i In the last decade much effort has been dedicated to the study of the magnetic coupling in nanoscale multilayer systems, because ultrathin magnetic films exhibit unusual magnetic configurations and coupling not found in bulk systems [1,2]. In particular, in trilayer systems like ferromagnet–nonmagnetic spa- cer–ferromagnet (e.g., Fe/Cr/Fe, Fe/Cu/Fe, etc.) long-periodic oscillating exchange coupling as a func- tion of spacer thickness has been found [3,4–7]. This means that the coupling changes from antiferromag- netic to ferromagnetic with spacer thickness. For deeper understanding of indirect exchange coupling through a nonmagnetic spacer one needs to investigate the magnetic properties of a nonmagnetic spacer in a multilayer system. Moreover, investigation of the magnetic properties of a nonmagnetic spacer sand- wiched between ferromagnetic layers is very impor- tant not only because of oscillating exchange cou- pling, but also because the contact of a ferromagnetic layer with a nonmagnetic spacer must change the elec- tronic state of the nonmagnetic spacer. Also, many ex- perimental and theoretical works are dedicated to the study of such trilayer systems as antiferromag- net–nonmagnetic spacer–antiferromagnet. Such sys- tems attract much interest of researchers because they can be used in magneto-resistive devices, for example, in magnetic field sensors, magnetic heads in memory devices, etc. Taking into account the aforementioned problems some works are dedicated to investigation of the distribution of magnetization induced in a non- magnetic metal spacer sandwiched between two ferro- magnetic layers. Of great interest is the question of the possibility of inducing magnetization in a material which in the normal state is nonmagnetic but which can be polarized if it boundaries with a magnetically ordered material [8]. Magnetic polarization of a non- magnetic material by a magnetically ordered material is usually called the proximity effect. The experimen- tal study of the proximity effect is described in some works [9–11]. For example, magnetic polarization of a nonmagnetic Au spacer in multilayer systems like ferromagnet–nonmagnetic spacer–ferromagnet has been measured with the help of M�ssbauer spectroscopy us- ing probe atoms in an Au spacer [12]. A small induced magnetic moment in Cu at the Co/Cu interface was detected Refs. 13,14 using circular dichroism and in [15] using NMR. The oscillating exchange coupling between two ferromagnetic layers separated by a non- © V.Yu. Gorobets, 2004 magnetic metal spacer is explained in some theoretical works, usually using RKKY coupling [16]. Moreover, oscillatory exchange coupling between two ferromag- netic layers separated by a nonmagnetic metal spacer layer is associated with oscillation of the magnetic mo- ment in the nonmagnetic spacer layer [3,17–19]. The magnetic moment induced in a nonmagnetic metallic spacer between two ferromagnets with magnetizations turned at an arbitrary angle is calculated theoretically in Ref. 17. As is shown in Ref. 17, the induced magne- tization rotates along a complex three-dimensional spiral and can undergo many complete rotations. In this work we propose a phenomenological method of calculating both the oscillating exchange coupling of magnetically ordered layers through a nonmagnetic metallic spacer and the induced magneti- zation in a the nonmagnetic spacer using a spin-den- sity model [20] similar to the Ginzburg–Landau mo- del [21]. In comparison with Ref.17 the approach proposed in our work allows to calculate the magneti- zation induced in the spacer and the oscillating ex- change coupling taking into account the influence of external magnetic field. The approach can be used whether the magnetically ordered layers are ferromag- netic or antiferromagnetic. To consider the system magnetically ordered layer–nonmagnetic metallic spacer–magnetically ordered layer (MOL–NMMS– MOL) in magnetic field, let us consider the case when external magnetic field H is parallel to the z axis and directed perpendicularly to the plane of the spacer. The interface of the spacer with the first MOL is situ- ated in the xy plane, the interface of the spacer with the second MOL is situated in the z L� plane parallel to xy. Following Ref. 20 let us define the order param- eter for the spacer as the two-component function � � � � �� 1 2 , (1) in terms of which the magnetic moment density of the spacer is written as M � � � �0 �σ , (2) where 0 is a phenomenological parameter and �σ are the Pauli matrices. We suppose in our approach that the state of the MOLs does not depend on distribu- tion of the order parameter inside the spacer and is characterized only by the directions of homogenous magnetization. Using the approach of Ref. 20 and the functional method of the Ginzburg–Landau type for the order parameter �, we write the following Lag- rangian function: L i w d� � �� 1 2 �( � � ) ( )� � � � � r. (3) Here w is the energy density written to forth-order accuracy in powers of the function �: w A s H z( ) ( ) ( � ),� � � �� 2 2 0 (4) where A, �, and s are phenomenological parameters of the spacer. From Eq. (3) in a trivial manner we obtain i t A s H z� � � � � � �� ( ) �0 (5) for the order-parameter function, which depends on time and coordinates inside the spacer. The boundary conditions have the form �� z � �0 const, �� z L� � const, (6) where the points z � 0 and z L� are the coordinates of the interfaces (see Fig. 1). The solution of Eq. (5) is sought in the form � � � � � � � � �� 1 1 2 2 ( ) ( ) r r exp ( ) exp ( ) i t i t . (7) By inserting Eq. (7) into Eq. (3) one obtains the system of nonlinear equations which defines the de- pendence of �1 and �2 on the space coordinates r: � � � � � � � � � � � � 1 1 1 1 2 2 2 1 1 2 2 2 1 2 0� � � � � � � � a b h a b (\| \| \| \| ) (\| \| \| 2 2 2 2 0\| ) � �� h , (8) where ai i� �(�� ; b s/A� ; h H/A� 0 . For simplicity of consideration we suppose that the external magnetic field H changes the directions of the magnetizations of the MOLs in the same way, i.e., the polar angle � � 0 0� H for the MOL magne- tizations is the same, and the difference in their orien- tation is described only by change of azimuth angle !. Let’s choose the constants in the boundary conditions (6) taking into account that the magnetic moment density of the spacer at the interfaces must be parallel or antiparallel, respectively, to the direction of mag- netization of the MOLs on variation of the sign of the exchange coupling. Also we suppose that exchange coupling between the magnetization in the spacer and the magnetizations of the MOLs is appreciable only within the interfacial regions (a L"" ), i.e., that MOLs influence the spacer only through the boundary conditions (6). Let’s consider that the magnetization in the spacer at the boundaries with the MOLs is par- allel to the magnetization of the MOLs. The case when the orientation is antiparallel can be considered analogously. In accordance with the above-said, one can write the boundary condition using a spherical co- 1046 Fizika Nizkikh Temperatur, 2004, v. 30, No. 10 V.Yu. Gorobets ordinate system (see Fig. 1) and Pauli matrices, in the form � � M i M z z 0 0 0 1 2 2 1 0 0 1 2 2 1 0 0 0 sin ( ) ( ) * * * * � � � � � � � �cos ( )* * 0 0 1 1 2 2 0 � � � � � � � � � � z , (9) �M M i z L 0 0 0 1 2 2 1 0 0 0 1 sin cos ( ) sin sin ( * * * ! ! � � � � � � � 2 2 1 0 0 0 1 1 2 2 � � � � � � � � � � � � * * * ) cos ( ) z L z L M , (10) where M0 is the magnitude of the magnetization at the boundaries, which plays the role of a phenomenological parameter of our problem. The magnitude \| \|M is chosen the same on both interfaces, equal to M /V0 0 0# (where V0 is the unit cell vol- ume of near the interface of a spacer layer of thick- ness a, V L L ax y0 � ), taking into account the assump- tion of the strong polarizing effect of MOLs on the magnetization of the spacer in the immediate neigh- borhood of the interface. It’s easy to see using direct substitution, that � � � � � � � � � � �� B B i i ik z i t i i ik z i t 1 2 1 1 1 1 2 2 2 2 e e e e e e e e κ ρ κ ρ � � � , (11) where κ i , $ i , and �i are real magnitudes, and ρ � ( , )x y satisfies the system of nonlinear equations (8) if the following correlations between parameters are fulfilled: � � � � � � � � � � � � ( ) ( ) ( ) ( % % 1 2 1 2 1 1 2 2 2 2 2 2 2 2 1 2 2 0k a b B B h k a b B B2 0) � � � � � �� h . (12) To make the solution (11) satisfy the boundary condi- tions, it is necessary that � � � $ $ ! 1 2 2 1 2 1 0 � � � � � � � � � � � κ κ κ , B m B m 1 0 0 2 0 0 2 2 � � � � �� cos sin , (13) k k N L1 2 2 � � �! & , where the conditions N � '0 1, , ..., m M 0 0 0 � (14) are fulfilled, too. Then the solution satisfying the sys- tem of nonlinear equations (8) and the boundary con- ditions (9), (10) is transformed to the form � � � ! & ! & � �� m i N L HL A N i t i 0 0 0 2 2 2 2 e e exp ( ) cos κρ � � ( ) � � � � � � � � � � � � � � � � �� ( ) z i N L HL A N sin ! & ! & 0 0 2 2 2 2 exp � � � � � � � � � � � � � � � � � � � � � � � � � � + , � ' z H N, , , , , ...if ! 0 0 0 1 � �� & & � �� ( ) * � � �� m i N L HL AN z i t i 0 0 0 2 2 e e exp κρ cos �� ( ) * � � �� sin & & 0 0 2 2 exp i N L HL AN z � � � � , � ' ', if ! 0 0 1 2, , , , ...H N (15) By inserting the solution (15) into the correlation (12) one obtains - � % ! & ! &N m s A A N L A HL N � � � � �� 0 2 2 1 0 2 2 2 2 , (16) Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetic layers Fizika Nizkikh Temperatur, 2004, v. 30, No. 10 1047 M M �� z y x � x y z L a b Fig. 1. Direction of magnetization of the spacer at the boundary with the first MOL, z = 0 (a) and with the boundary with the second MOL, z = L (b). where - �N � � . Let us assume, that Eq. (15) de- scribes the states of quasiparticles (which we will call magnetized electrons), which behave according to Fermi–Dirac statistics. According to Ref. 22 let us define the total number of magnetized electrons in the spacer magnetopolarized by the MOLs, N0 as a sum over all possible states of the distribution function. Let’s consider, for simplicity, the case when T � 0, because the most interesting magnetoresistive proper- ties of multilayered nanostructured systems appear at low temperatures. Then the Fermi–Dirac distribution function has the form f N N F N F 0 1 0 ( ) , , - - - - - � . , � � � , (17) where -F is the Fermi energy of the magnetized elec- trons. According to Eq. (17), the total number of mag- netized electrons in the spacer is written in the form N L L d x y N N 0 2 0 2 2 2� �� ( )& &% % � , (18) where L L Lx y# ,, are the dimensions of the spacer in the xy plane, and %N is the largest possible value of the wave vector κ when N is fixed. κ is calculated as % - � ! & ! &N F m s A N L HL A N � � � � �� 0 2 0 2 2 2 2( ) . (19) It is easy to obtain from Eq. (13) that nL N N � �1 2 2 & % , (20) where n N / L L Lx y� 0 ( ) is the density of magnetized electrons in the spacer. The components of the spe- cific magnetization of magnetized electrons in a plane with an arbitrary coordinate z inside the spacer, tak- ing into account Eq. (2), have the form: M M N L z M M N L z xN yN � �� 0 0 0 0 2 2 sin co sin sin ! & ! & s � � � � � � � � � � � � � � M MzN 0 0cos . (21) The average x component of the specific magnetiza- tion of the near-interface spacer layer of thickness a equals / 0 � �� M L L M N L z dzxN x y a 0 0 0 2 sin cos ! & � �� LL L M N L a N x y 0 0 2 2 sin sin ! & ! & . (22) The average y component of the specific magnetiza- tion of the near-interface spacer layer of thickness a equals / 0 � �� M L L M N L z dzyN x y a 0 0 0 2 sin sin ! & � � �� LL L M N L a N x y 0 0 2 2 sin cos ! & ! & . (23) The average z component of the specific magnetiza- tion of the near-interface spacer layer of thickness a accordingly equals / 0 �M L L M azN x y 0 0cos . (24) In the limiting case when a L"" , the average absolute value of the magnetization of near-interface spacer layer, / 0 � / 0 � / 0 � / 0M M M MxN yN zN 2 2 2 , equals L L M ax y 0 . With the help of Eq. (17) one can find the magnetization of the near-interface spacer layer of thickness a: M L L M d x y N N � ��2 2 2 0 2 0 ( )& &% % � . (25) Taking into account Eq. (20), one can find that M M nL L Lx y� 0 . Using the ratio M / L L ax y0 0� ( ), one can write M nL a � 0 , (26) hence: n Ma L � 0 . (27) We denote n M/0 0� , n a /L� 0 , and a Ma/0 0� is a phenomenological constant. Taking into account Eqs. (19) and (20), one ob- tains a transcendental algebraic equation for finding the Fermi energy -F : nL m s A N L F N � � � � �� 1 2 2 2 0 2 & - � ! & � � � � �� ( ) � � ! & 0 2 2 HL A N( ) , (28) where the allowed values of N are found from the in- equality %N 2 01 . N belongs to 1048 Fizika Nizkikh Temperatur, 2004, v. 30, No. 10 V.Yu. Gorobets N 2 � � � � � ( ) � � � � � ( ) * � � � � � 3floor ceil c ! & & ! & &2 2 2 2 2 1; eil floor� � ( ) * � � � � � � � � ( ) * � � � � � ! & & ! & &2 2 2 2 1 2; , (29) where - � - � 1 0 0 2 0 2 2� � � � � �� L m s A m s A H A F F , (30) - � - � 2 0 0 2 0 2 2� � � � � �� L m s A m s A H A F F , (31) ceil is the function that gives the maximal integer number closest to a given real number, and floor is the function that gives the minimal integer number closest to a given real number. The average energy of the spacer is defined by the standard formula / 0 � ��- & &%- % % � 2 2 2 2 0 L L d x y N N N ( ) ( ) . (32) After simple transformations, Eq. (32) transforms to: / 0 � � � - � L L m s nL x y ( )0 � � �� A m s A N L F N 4 2 2 0 2 4 & - � ! & � � � � �� ( ) � � ! & 0 4 0 2 2 1 2 HL A N H A( ) , (33) where / 0- / L Lx y( ) is the spacer’s energy per a unit area. By comparing the experimental data for the ex- change coupling energy between MOLs and the maxi- mal exchange coupling energy between the layers [23–29] for different types of trilayer systems with the results of numerical calculations using the given mo- del, it was found that if the density of magnetized electrons n equals the tabulated point for the cor- responding metal spacer, then the parameter a # 1 � and the parameter A may occupy the region 10 1030 2 27 2� �4 " " 4erg cm erg cmA . For example, Eq. (28) was solved numerically for the following val- ues of the spacer’s parameters: A � 4 4�015 10 30 2. erg cm , n � 4 �5 9 1022 3, cm , & � 50 � , a � 1 �. The curves of the Fermi energy as a function of the spacer’s thick- ness, on magnetic field, and on the angle !, while the remaining parameters are fixed, are shown, are per- formed in Fig. 2, 3, and 4, respectively. It is obvious from the plots that the Fermi energy has oscillating dependence on the spacer’s thickness and on magnetic field. These oscillations are a purely dimensional ef- fect analogous to that described Ref. 22. The results of numerical calculation of / 0- / L Lx y( ) for the spacer’s parameters, taking into account calcu- lated numerically Fermi energy, are shown in Figs. 5 Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetic layers Fizika Nizkikh Temperatur, 2004, v. 30, No. 10 1049 0 0.2 0.4 0.6 0.8 1.0 0.080 0.085 0.090 0.095 0.100 0.105 � � � / Fig. 2. Dependence of Fermi energy on the angle �. Da- shed line H � 800 Oe, dotted line H � 400 Oe, solid line H � 0 Oe. � �0 2 2 33� A N /( ) , � � �� F m s0 , L = 23 �. 0 5 10 15 20 25 30 35 40 45 0.05 0.10 0.15 0.20 0.25 0.30 L, / Fig. 3. Dependence of Fermi energy on the spacer’s thick- ness. Solid line � � 0, dashed line � �� , H = 0. and 6. It is obvious from the Figs. 5 and 6 that both / 0- / L Lx y( ) and -F also have oscillating behavior de- pending on the spacer’s thickness and magnetic field. / 0- / L Lx y( ) and -F also have minima when ! � 0 or ! &� , depending on spacer thickness, magnetic field, and other parameters. That is, the energy-optimal con- figuration is the parallel or antiparallel mutual orien- tation of the projections of the magnetizations of the MOLs on the xy plane. As is seen from Fig. 7, the exchange coupling of the MOLs through the metal spacer has a long-periodic oscillatory dependence on the spacer’s thickness L, i.e., the exchange coupling changes from antifer- romagnetic to ferromagnetic depending on the spacer’s thickness. As is seen from Fig. 5, the given theore- tical calculation shows that �J / L Lx y� / 0 � � - � ( ) �� / 0 � - � / L Lx y( ) 0 descends faster than L�2 with in- crease of L. Moreover, the period of the oscillations of J is not constant but increases with increase of L. Both of these theoretical results agree quantitatively with results of Ref. 29 (see Fig. 5). As is seen from Figs. 8 and 9, the theoretically obtained values of the first three periods of oscillation of the exchange coupling equal 10, 12, and 15 �, respectively. The experimen- tally obtained periods of oscillation of the exchange coupling (according to Ref. 29) equal to 11.0, 13.6 and 15.2 � with accuracy ±1 �. The components of the magnetization at a point with an arbitrary coordinate z inside the spacer have the form (21). Then, taking into account the Fer- mi–Dirac statistics and the distribution functions (17), the mean value of the magnetization components 1050 Fizika Nizkikh Temperatur, 2004, v. 30, No. 10 V.Yu. Gorobets 0 5 10 15 20 0.08 0.10 0.12 0.14 0.16 0.18 H, kOe / Fig. 4. Dependence of the Fermi energy on magnetic field. Solid line � � 0, dotted line � �� , L = 23 �. 10 20 30 40 50 –20 –15 –10 –5 0 5 L,� J, 1 0 e rg /c m – 3 2 Fig. 5. Dependence of exchange coupling energy � �J / L L / L Lx y x y� � � � � � � � � � ( ) ( ) 0 on the spacer’s thickness. Solid line is theoretical calculation, «o» are the results of the experimental work [29]. It was found by comparing these theoretical and experimental results that they are shifted by 6 �. 0 5 10 15 20 –25 –20 –15 –10 –5 0 5 10 15 20 H, kOe J, 1 0 e rg /c m – 3 2 Fig. 6. Dependence of the exchange coupling energy J on magnetic field. L = 23 �. 0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1.0 L, � � m in / Fig. 7. Dependence of the angle when the energy of cou- pling of the MOLs is minimal as a function of the spacer’s thickness. H � 0. at an arbitrary point with coordinate z inside the spacer has the form: / 0 � / 0 � ��M L L d M M L L d M x x y xN N y x y N 2 2 2 2 2 2 2 0 2 ( ) ( ) & &% % & &% % � yN N z x y zN N N N M L L d M 0 2 0 2 2 2 � � & &% % �� / 0 � � � � � � � � � � � � � � ( ) . (34) It is simple to transform Eq. (34) to the form: / 0 � �� 6�M L L M N L zx x y N 1 2 2 0 0& ! & sin cos � � 6 � � � �� - � ! & ! & F m s A N L HL A N 0 2 0 2 2 2 2 ( ) � � , (35) / 0 � �� 6� M L L M N L z y x y N 1 2 2 0 0& ! & sin sin � � 6 � � � �� - � ! & ! & F m s A N L HL A N 0 2 0 2 2 2 2 ( ) � *� , (36) / 0 � M L L M nLz x y 0 0cos . (37) The absolute value of the magnetization at a point with coordinate z inside the spacer is calculated as � �/ 0 � / 0 � / 0 � / 0M M M Mx y z 2 2 2 . (38) The plots of the dependences of / 0Mx and / 0My on coordinate z inside the spacer are performed in Figs. 8 and 9. Here the boundary conditions are chosen so that when z � 0, the magnetization of the spacer is di- rected along the x axis, and when z L� the magne- tization of the spacer is directed along the y axis. The magnitude / 0Mz , as is seen from Eq. (37), is con- stant at all spacer thicknesses, according to our mo- del. As seen from Fig. 10, the average value of � �M is maximal on the spacer’s boundaries with the MOLs and decays from the spacer’s boundaries inside the spacer symmetrically about of the spacer middl. Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetic layers Fizika Nizkikh Temperatur, 2004, v. 30, No. 10 1051 0 0.2 0.4 0.6 0.8 1.0 –0.2 0 0.2 0.4 0.6 0.8 1.0 z/L M x Fig. 8. Magnetization component of Mx (divided by the magnetization at the z � 0 boundary) as a function of z. Solid line H � 0 Oe, dotted line H � 2000 Oe. The spacer’s thickness is 23 �. 0 0.2 0.4 0.6 0.8 1.0 –0.2 0 0.2 0.4 0.6 0.8 1.0 z/L M y Fig. 9. Magnetization component of My (divided by mag- netization at the z L� boundary) as a function of z. Solid line H � 0 Oe, dotted line H � 2000 Oe. The H � 2000 Oe spacer’s thickness is 23 �. 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 z/L \|M \| Fig. 10. Absolute value of M (divided by the magnetiza- tion at the z � 0 boundary) as a function of z. Solid line H � 0 Oe, dotted line H � 2000 Oe. The spacer’s thickness is 23 �. Discussion of results In this paper the exchange coupling of magneti- cally ordered layers through a nonmagnetic metallic spacer was calculated using a spin-density model simi- lar to the Ginzburg–Landau model. The induced mag- netization in the spacer, taking into account the influ- ence of an external magnetic field, was calculated, too. This calculation shows that the energy of cou- pling of the MOLs through the nonmagnetic metallic spacer is an oscillating function of the spacer’s thick- ness and magnetic field, i.e., the exchange coupling between the layers varies from ferromagnetic to anti- ferromagnetic and vice versa in variation of the spacer’s thickness and magnetic field. Here the magni- tude of the exchange coupling decreases with spacer thickness faster than L�2 and the period of oscillation of the exchange coupling is not constant but increae- ses with increase of spacer thickness. These results match with the works [29–31], which were carried out for Cu and Au spacers and for Co and Fe MOLs. Also, this calculation shows that in a nonferromagnetic spacer the induced magnetization can undergo many complete rotations with variation of distance to the boundaries with the MOLs. Moreover, the absolute value of the induced magnetization decays nonmo- notonically with distance from the interfaces inside the spacer. It is shown that the character of the decay of the absolute value of the magnetization from the in- terfaces into the interior of the spacer is influenced by magnetic field. The author is grateful to Prof. V.G. Baryakhtar for fruitful discussion of the results and to Prof. A.V. Svidzinsky for discussion of basic aspects of the model used in this work for calculation of the induced mag- netization of the spacer. 1. S.S.P. Parkin, Phys. Rev. Lett. 71, 1641 (1993). 2. P. Bruno, Phys. Rev. B52, 411 (1995). 3. S.S.P. Parkin, N. More, and K.P. Roche, Phys. Rev. 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Magnetization of a nonferromagnetic metal spacer sandwiched between two magnetically ordered layers

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