- Email: [email protected]
Magnetic resonance frequencies in multilayers with biquadratic exchange and non-collinear magnetic ordering
Journal of Magnetism and Magnetic Materials 165 (1997) 468-470
~ l ~ Journalof magnetism and magnetic ~ l ~ materials
ELSEVIER
Magnetic resonance frequencies in multilayers with biquadratic exchange and non-collinear magnetic ordering N.G. Bebenin *, A.V. Kobelev, A.P. Tankeyev, V.V. Ustinov Institute of Metal Physics, Ural Division of the Russian Academy of Sciences, S. Kovalevskaya Str. 18, 620219 Ekaterinburg, GSP-170, Russia Abstract
Magnetic resonance frequencies in a magnetic trilayer or superlattice are theoretically studied. Biquadratic exchange between layers is assumed to be sufficiently strong for non-collinear (canted) magnetic ordering to exist in the multilayer. The optic mode frequency turns out to vanish at the boundary between antiferromagnetic and canted states. It is shown that in the case of longitudinal pumping one can observe the resonance at two different values of magnetic field strength.
Keywords: Multilayers; Exchange coupling - biquadratic; Ferromagnetic resonance
superlattice per unit area and per one magnetic cell can be written as
I. Introduction
Ferromagnetic resonance (FMR) is known to be a good tool to investigate interactions in magnetic multilayers. For example, the FMR was used for studying the exchange coupling between magnetic layers in F e / C r [i], permall o y / C u [2], C o / R u [3] and other systems, in the theory used in those papers, the exchange interaction was taken bilinear in magnetizations of the magnetic layers. It has been, however, demonstrated that biquadratic exchange can play an important role in trilayers, see Refs. [4,5] and references therein, as well as in superlattices [6-8]. Therefore it is necessary to develop the theory in which the biquadratic exchange is taken into account. 2. Equations of motion
Let us consider a trilayer or a superlattice consisting of alternating magnetic (of thickness d m) and non-magnetic layers. Suppose that the superlattice can be treated as a sum of two magnetic sublattices. Assume that magnetizations of the magnetic layers of the trilayer are Mj and M 2, IM~I = In21 =Mm; if the superlattice is considered, M 1 ( M z) refers to the magnetization inside a magnetic layer that belongs to the first (second) sublattice. The free energy of the trilayer per unit area or the free energy of the
* Corresponding author. Fax: [email protected].
+ 7-3432-445244;
email:
( M , M2) F =
-
Jl
- K1
Z
Mm
J2
(MIM2) 2 4 Mm
( n M ! ) 2 + (riM2) 2 2 + L!
Mm
(nM!)(nM2) 2
Mm
+ 2"n'dm[(nM,) 2 + ( n M 2 ) 2] - d m H ( M 1 + M 2 ) .
(i) The first two terms of (1) are bilinear and biquadratic exchange energy, respectively, the third describes the uniaxial anisotropy inside a magnetic layer, n is the normal to the layers plane, the fourth is the anisotropy part of interlayer interaction. The last two terms are demagnetization and Zeeman energy of the film in a uniform external magnetic field H. We take into account only one term of the fourth order, namely, the biquadratic exchange, which is isotropic and hence the most important; the role of the anisotropic terms of the fourth order will be discussed elsewhere. It is assumed that in the absence of a magnetic field the magnetic moments lie in the film plane. The magnetic ordering of the structure is ferromagnetic (FM) if Jl > 0, J1 + 2 J2 > 0, or antiferromagnetic (AFM) when Jl < 0, Jl - 2 J2 < 0, or non-collinear (NC) that takes place in the event J1 - 2 J2 > 0, Jl + 2 J2 < 0. The phase diagram of a multilayer is shown in Fig. 1. Neglecting dissipation, one can write the equation of motion for M 1 as i OM1 y Bt [ M , XH~ff], (2)
0304-8853/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. Pll S0304-8 853(96)00594-X
N.G. Bebenin et al./ Journal of Magnetism and Magnetic Materials 165 (1997) 468-470
AFM
the equation for M allows to obtain a magnetization curve, i.e. dependence of the relative (with respect to the saturation value) magnetization m on magnetic field. We assume that H o lies in the film plane and set n = (0,0,1), H o = (0,H o,0), M o = (0,M o,0), L o = (L o,0,0). One can show easily that because of the biquadratic exchange a magnetization curve is not a straight line, see Fig. 2. In the NC phase, the equation for m ( H o) can be written as
FM
.1, 3
469
8m3Hez + 2rn(Hel - 2He2 ) + H o = 0.
Nc
(5)
It follows from Eq. (5) that the zero-field magnetization, m o = m ( H o = 0) and the saturation field, H~, are given by Fig. 1. Phase diagram.
}/n~l - 2/-/°2 mo =
-4He2
, H s = - 2 ( H e , + 2He2 ).
(6)
where OF
H~ ff =
M2
dmC3M 1
H+
n(nMl) --HKI
- -
Mm
nelMmm
+ 2He2
3. Resonance frequencies
( M 1M 2) M 2 3 Mm
Hz, n ( ~ 2) JVlm '
(3)
J2
J1 =
/'/el = dmM--'-'~, 2K1
/4K1 = 4~Mm
dmM m '
LI HL, =
dmM m '
(4)
and 3' is gyromagnetic ratio. As is evident from Eq. (3), the effective field acting on M I depends not only on M 2 but also on the angle between M 1 and M s. The equation for M z can be obtained from Eqs. (2) and (3) by substituting M 2 for M 1 and vice versa. It is convenient to introduce the variables M = M~ + M 2 and L = M 1 - M 2 the equations for which follow from Eq. (2). In the case of static magnetic field, H = H o,
Now let us suppose that the weak radio frequency field is applied. Writing M ( t ) = M o + ~ M e -i'°t, L ( t ) = L o + 8 L e -i'°t, H ( t ) = H o + h e -i'°t, where ~o is the frequency, after linearization of the equations of motion with respect to 8M, ~L, and h we obtain the system of six equations for components of 8 M and 8L. In the AFM state over the range 0 < H o < H s, when both M o and L o are not equal to zero, the set of equations splits into two separate systems one of which describes 8 M x, ,~My and ~)Ly in the rf field with h x :g 0 and (or) h z -4:0 whereas the second describes ,~My, 8 L x and 8 L z in the field h II Ho. Equating the determinant of the first system to zero, one finds the frequency of the in-phase motion of M I and M 2 ('acoustic' mode): w I = y~Ho( H o + m(H~: 1 - H L , ) ) .
(7)
In the same way one obtains the frequency of the out-of phase ('optic') mode by making use of the second system. The result is
1 oo2=y~[/(Hxl+HL,)(1-m)(Ho-8He2m3).
0q HJH s
Fig. 2. Schematic plot of magnetization curves of a multilayer in different magnetic states. The numbers labeling the curves correspond to the points on the phase diagram shown in Fig. 1.
(8)
It is easy to see that w 1 = 0 for H o = 0 and ~o1 increases with increasing field. The optic mode frequency does not generally vanish when H o --* 0 but always oJ2 = 0 if H o = H s. In an ordinary anfiferromagnet, oJ2(H o) is a monotone decreasing function. If the biquadratic exchange is strong enough, ~o2(Ho) may be non-monotonic, see Fig. 3. At the boundary between the AFM and NC phases, i.e. when Jl < O, J1 - 2 Jz = 0, this function vanishes both at H o = 0 and H o = H s and hence has a maximum at a point which lies between zero and Hs. It is easy to show that to2(H o) has a minimum at H o = 0 and a maximum at an internal point of the interval (0, H o) not only at this phase bound-
N.G. Bebenin et al. / Journal of Magnetism and Magnetic Materials 165 (1997) 468-470
470
excited by a uniform rf field. It is easy to see that to 1 turns into to 3 and to 2 turns into to4 at H 0 = H s.
4. Conclusion
¢
ItJH, Fig. 3. Sketch of the magnetic field dependence of to2 for the different types of magnetic ordering. The notations are the same as in Fig. 2.
ary but also every time when the inequality H~l -- 10He2 > 0 is satisfied. In the NC phase the minimum at H o = 0 exists if 3Hel + 2H~2 < 0. One can rewrite this inequality as m 2 < 2 / 3 ; in other words, the point on the phase diagram must be far enough from the boundary between the NC and F M phases. Let us proceed to the consideration of the states with L 0 = 0 that takes place in the FM phase or when the constant magnetic field exceeds the saturation value. The set of the equations for 8 M and 8 L splits into two systems for ~ M x, ~ M z and ~L x, 8 L z, respectively, whereas ~My = ~L r = 0. Thus there are two modes with the frequencies ,o3 = ~g/Ho(H o + ~ , . - H ~ , ) .
to4= T~/( n o -
n s ) ( n o - ns + n g l + HL, ) .
(9) (10)
The first mode can be excited by a rf field with non-zero h , and(or) h z while the second one cannot be
The formulas for the magnetic resonance frequencies in a magnetic multilayer differ significantly from those for an ordinarily antiferromagnet if the biquadratic exchange is strong and especially if the non-collinear magnetic state exists. The longitudinal excitation is most interesting because in this case firstly the biquadratic exchange appears in the resonance frequency expression in explicit form and secondly the magnetic resonance can occur at two different values of magnetic field strength if the proper frequency of the external rf field is used. Acknowledgement. The research described in this publication was made possible by Grant No. 95-02-04813 from the Russian Foundation of Basic Researches.
References [1] J.J. Krebs, P. Lubitz, A. Chaiken and G.A. Prinz, Phys. Rev. Lett. 63 (1989) 1645. [2] A.Z. Maksymowicz, J.S.S. Whiting, M.L. Watson and A. Chambers, J. Magn. Magn. Mater. 94 (1991) 109. [3] Z. Zhang, L. Zhou, P.E. Wigen and K. Ounadjela, Phys. Rev. B 50 (1994) 6094. [4] Q. Leng, V. Cros, R. Schafer, A. Fuss, P. Griinberg and W. Zinn, J. Magn. Magn. Mater. 126 (1993) 367. [5] J.C. Slonczewski, J. Magn. Magn. Mater. 150 (1995) 13. [6] A. Schreyer, J. Fankner, M. Schafer, Th. Zeidler, H. Zabel, C.F. Majkrzak and P. Griinberg, J. Magn. Magn. Mater. 148 (1995) 189. [7] V.V. Ustinov, M.M. Kirillova, I.D. Lobov, V.M. Mayevskii, A.A. Makhnev, V.I. Minin, L.N. Romashev, A.R. Del, A.V. Semerikov and E.I. Shreder, JETP 109 (1996) 1 (in Russian). [8] V.V. Ustinov, L.N. Romashev, V.I. Minin, A.V. Semerikov and A.R. Del, Fiz. Met. Metalloved. 80 (1995) 71 (in Russian).
~ l ~ Journalof magnetism and magnetic ~ l ~ materials
ELSEVIER
Magnetic resonance frequencies in multilayers with biquadratic exchange and non-collinear magnetic ordering N.G. Bebenin *, A.V. Kobelev, A.P. Tankeyev, V.V. Ustinov Institute of Metal Physics, Ural Division of the Russian Academy of Sciences, S. Kovalevskaya Str. 18, 620219 Ekaterinburg, GSP-170, Russia Abstract
Magnetic resonance frequencies in a magnetic trilayer or superlattice are theoretically studied. Biquadratic exchange between layers is assumed to be sufficiently strong for non-collinear (canted) magnetic ordering to exist in the multilayer. The optic mode frequency turns out to vanish at the boundary between antiferromagnetic and canted states. It is shown that in the case of longitudinal pumping one can observe the resonance at two different values of magnetic field strength.
Keywords: Multilayers; Exchange coupling - biquadratic; Ferromagnetic resonance
superlattice per unit area and per one magnetic cell can be written as
I. Introduction
Ferromagnetic resonance (FMR) is known to be a good tool to investigate interactions in magnetic multilayers. For example, the FMR was used for studying the exchange coupling between magnetic layers in F e / C r [i], permall o y / C u [2], C o / R u [3] and other systems, in the theory used in those papers, the exchange interaction was taken bilinear in magnetizations of the magnetic layers. It has been, however, demonstrated that biquadratic exchange can play an important role in trilayers, see Refs. [4,5] and references therein, as well as in superlattices [6-8]. Therefore it is necessary to develop the theory in which the biquadratic exchange is taken into account. 2. Equations of motion
Let us consider a trilayer or a superlattice consisting of alternating magnetic (of thickness d m) and non-magnetic layers. Suppose that the superlattice can be treated as a sum of two magnetic sublattices. Assume that magnetizations of the magnetic layers of the trilayer are Mj and M 2, IM~I = In21 =Mm; if the superlattice is considered, M 1 ( M z) refers to the magnetization inside a magnetic layer that belongs to the first (second) sublattice. The free energy of the trilayer per unit area or the free energy of the
* Corresponding author. Fax: [email protected].
+ 7-3432-445244;
email:
( M , M2) F =
-
Jl
- K1
Z
Mm
J2
(MIM2) 2 4 Mm
( n M ! ) 2 + (riM2) 2 2 + L!
Mm
(nM!)(nM2) 2
Mm
+ 2"n'dm[(nM,) 2 + ( n M 2 ) 2] - d m H ( M 1 + M 2 ) .
(i) The first two terms of (1) are bilinear and biquadratic exchange energy, respectively, the third describes the uniaxial anisotropy inside a magnetic layer, n is the normal to the layers plane, the fourth is the anisotropy part of interlayer interaction. The last two terms are demagnetization and Zeeman energy of the film in a uniform external magnetic field H. We take into account only one term of the fourth order, namely, the biquadratic exchange, which is isotropic and hence the most important; the role of the anisotropic terms of the fourth order will be discussed elsewhere. It is assumed that in the absence of a magnetic field the magnetic moments lie in the film plane. The magnetic ordering of the structure is ferromagnetic (FM) if Jl > 0, J1 + 2 J2 > 0, or antiferromagnetic (AFM) when Jl < 0, Jl - 2 J2 < 0, or non-collinear (NC) that takes place in the event J1 - 2 J2 > 0, Jl + 2 J2 < 0. The phase diagram of a multilayer is shown in Fig. 1. Neglecting dissipation, one can write the equation of motion for M 1 as i OM1 y Bt [ M , XH~ff], (2)
0304-8853/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. Pll S0304-8 853(96)00594-X
N.G. Bebenin et al./ Journal of Magnetism and Magnetic Materials 165 (1997) 468-470
AFM
the equation for M allows to obtain a magnetization curve, i.e. dependence of the relative (with respect to the saturation value) magnetization m on magnetic field. We assume that H o lies in the film plane and set n = (0,0,1), H o = (0,H o,0), M o = (0,M o,0), L o = (L o,0,0). One can show easily that because of the biquadratic exchange a magnetization curve is not a straight line, see Fig. 2. In the NC phase, the equation for m ( H o) can be written as
FM
.1, 3
469
8m3Hez + 2rn(Hel - 2He2 ) + H o = 0.
Nc
(5)
It follows from Eq. (5) that the zero-field magnetization, m o = m ( H o = 0) and the saturation field, H~, are given by Fig. 1. Phase diagram.
}/n~l - 2/-/°2 mo =
-4He2
, H s = - 2 ( H e , + 2He2 ).
(6)
where OF
H~ ff =
M2
dmC3M 1
H+
n(nMl) --HKI
- -
Mm
nelMmm
+ 2He2
3. Resonance frequencies
( M 1M 2) M 2 3 Mm
Hz, n ( ~ 2) JVlm '
(3)
J2
J1 =
/'/el = dmM--'-'~, 2K1
/4K1 = 4~Mm
dmM m '
LI HL, =
dmM m '
(4)
and 3' is gyromagnetic ratio. As is evident from Eq. (3), the effective field acting on M I depends not only on M 2 but also on the angle between M 1 and M s. The equation for M z can be obtained from Eqs. (2) and (3) by substituting M 2 for M 1 and vice versa. It is convenient to introduce the variables M = M~ + M 2 and L = M 1 - M 2 the equations for which follow from Eq. (2). In the case of static magnetic field, H = H o,
Now let us suppose that the weak radio frequency field is applied. Writing M ( t ) = M o + ~ M e -i'°t, L ( t ) = L o + 8 L e -i'°t, H ( t ) = H o + h e -i'°t, where ~o is the frequency, after linearization of the equations of motion with respect to 8M, ~L, and h we obtain the system of six equations for components of 8 M and 8L. In the AFM state over the range 0 < H o < H s, when both M o and L o are not equal to zero, the set of equations splits into two separate systems one of which describes 8 M x, ,~My and ~)Ly in the rf field with h x :g 0 and (or) h z -4:0 whereas the second describes ,~My, 8 L x and 8 L z in the field h II Ho. Equating the determinant of the first system to zero, one finds the frequency of the in-phase motion of M I and M 2 ('acoustic' mode): w I = y~Ho( H o + m(H~: 1 - H L , ) ) .
(7)
In the same way one obtains the frequency of the out-of phase ('optic') mode by making use of the second system. The result is
1 oo2=y~[/(Hxl+HL,)(1-m)(Ho-8He2m3).
0q HJH s
Fig. 2. Schematic plot of magnetization curves of a multilayer in different magnetic states. The numbers labeling the curves correspond to the points on the phase diagram shown in Fig. 1.
(8)
It is easy to see that w 1 = 0 for H o = 0 and ~o1 increases with increasing field. The optic mode frequency does not generally vanish when H o --* 0 but always oJ2 = 0 if H o = H s. In an ordinary anfiferromagnet, oJ2(H o) is a monotone decreasing function. If the biquadratic exchange is strong enough, ~o2(Ho) may be non-monotonic, see Fig. 3. At the boundary between the AFM and NC phases, i.e. when Jl < O, J1 - 2 Jz = 0, this function vanishes both at H o = 0 and H o = H s and hence has a maximum at a point which lies between zero and Hs. It is easy to show that to2(H o) has a minimum at H o = 0 and a maximum at an internal point of the interval (0, H o) not only at this phase bound-
N.G. Bebenin et al. / Journal of Magnetism and Magnetic Materials 165 (1997) 468-470
470
excited by a uniform rf field. It is easy to see that to 1 turns into to 3 and to 2 turns into to4 at H 0 = H s.
4. Conclusion
¢
ItJH, Fig. 3. Sketch of the magnetic field dependence of to2 for the different types of magnetic ordering. The notations are the same as in Fig. 2.
ary but also every time when the inequality H~l -- 10He2 > 0 is satisfied. In the NC phase the minimum at H o = 0 exists if 3Hel + 2H~2 < 0. One can rewrite this inequality as m 2 < 2 / 3 ; in other words, the point on the phase diagram must be far enough from the boundary between the NC and F M phases. Let us proceed to the consideration of the states with L 0 = 0 that takes place in the FM phase or when the constant magnetic field exceeds the saturation value. The set of the equations for 8 M and 8 L splits into two systems for ~ M x, ~ M z and ~L x, 8 L z, respectively, whereas ~My = ~L r = 0. Thus there are two modes with the frequencies ,o3 = ~g/Ho(H o + ~ , . - H ~ , ) .
to4= T~/( n o -
n s ) ( n o - ns + n g l + HL, ) .
(9) (10)
The first mode can be excited by a rf field with non-zero h , and(or) h z while the second one cannot be
The formulas for the magnetic resonance frequencies in a magnetic multilayer differ significantly from those for an ordinarily antiferromagnet if the biquadratic exchange is strong and especially if the non-collinear magnetic state exists. The longitudinal excitation is most interesting because in this case firstly the biquadratic exchange appears in the resonance frequency expression in explicit form and secondly the magnetic resonance can occur at two different values of magnetic field strength if the proper frequency of the external rf field is used. Acknowledgement. The research described in this publication was made possible by Grant No. 95-02-04813 from the Russian Foundation of Basic Researches.
References [1] J.J. Krebs, P. Lubitz, A. Chaiken and G.A. Prinz, Phys. Rev. Lett. 63 (1989) 1645. [2] A.Z. Maksymowicz, J.S.S. Whiting, M.L. Watson and A. Chambers, J. Magn. Magn. Mater. 94 (1991) 109. [3] Z. Zhang, L. Zhou, P.E. Wigen and K. Ounadjela, Phys. Rev. B 50 (1994) 6094. [4] Q. Leng, V. Cros, R. Schafer, A. Fuss, P. Griinberg and W. Zinn, J. Magn. Magn. Mater. 126 (1993) 367. [5] J.C. Slonczewski, J. Magn. Magn. Mater. 150 (1995) 13. [6] A. Schreyer, J. Fankner, M. Schafer, Th. Zeidler, H. Zabel, C.F. Majkrzak and P. Griinberg, J. Magn. Magn. Mater. 148 (1995) 189. [7] V.V. Ustinov, M.M. Kirillova, I.D. Lobov, V.M. Mayevskii, A.A. Makhnev, V.I. Minin, L.N. Romashev, A.R. Del, A.V. Semerikov and E.I. Shreder, JETP 109 (1996) 1 (in Russian). [8] V.V. Ustinov, L.N. Romashev, V.I. Minin, A.V. Semerikov and A.R. Del, Fiz. Met. Metalloved. 80 (1995) 71 (in Russian).