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Magnon modes in permalloy nanorings
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 286 (2005) 366–369 www.elsevier.com/locate/jmmm
Magnon modes in permalloy nanorings C.E. Zaspela,, B.A. Ivanovb a
Department of Physics, Montana State University, Bozeman, MT 59717, USA b Institute of Magnetism, NASU, 36B Vernadskii Ave. 03142 Kiev, Ukraine Available online 4 November 2004
Abstract Submicron permalloy rings can have a vortex ground state as a result of the competing effects arising from the exchange and magnetostatic interactions. The dynamic properties of this state are investigated by analytical calculations of the structure and frequencies of the magnon normal modes in the presence of the vortex–magnon interaction. For thin rings of thickness, L ¼ 60 nm and inner and outer radii, Ri and Ro=200 nm, respectively it is shown that the magnon node frequencies in the 10 GHz range for rings of radii of a few hundred nanometers. Moreover, the frequencies are slowly increasing functions of Ro for fixed Ro. r 2004 Published by Elsevier B.V. PACS: 75.30.Ds; 75.40.Gb; 75.75.+a Keywords: Spin waves; Magnetic vortex; Permalloy rings
Magnetic microdots made from a soft ferromagnetic material such as permalloy have been proposed [1] for use as magnetic storage media, and the study of the basic physics of magnetic particles in this size range has led to interesting effects. For example, when the dot radius is reduced to the micron and submicron range the competition between exchange and the magnetostatic interaction will result in a vortex ground state with the magnetization mainly confined to the dot plane. This structure is stable because the Corresponding author. Tel: 406 994 3614; fax: 406 994 4452.
E-mail address: [email protected] (C.E. Zaspel). 0304-8853/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.jmmm.2004.09.093
curling effect in the planar vortex will eliminate effective magnetostatic charges thereby minimizing the energy. However, at the core of the planar vortex there is a ‘energy singularity, that is eliminated by an out of plane rotation of the magnetization at the core resulting in a small surface charge, which has also been observed [2] in thin dots. In materials such as permalloy the presence of the core results in a net magnetic moment perpendicular to the dot plane over a central region of about 10 nm diameter. If arrays of dots are to be used in magnetic information storage, this vortex core presents several disadvantages. The net magnetic moment of the dot will
ARTICLE IN PRESS C.E. Zaspel, B.A. Ivanov / Journal of Magnetism and Magnetic Materials 286 (2005) 366–369
interact through the dipolar interaction with other dots in the array affecting properties such as switching of the curling direction. Because of the presence of the core, there is a low-frequency (sub gigahertz) precessional mode [3] that will complicate the dynamics of the single dot as well as the array of dots. Finally, the vortex state is only stable above a critical diameter of approximately 100 nm limiting the possible bit density in arrays of these systems. Some of these problems can be overcome by the use of a ring rather than a dot. The ring has no core, and therefore no magnetostatic fields other than those from fluctuations about the vortex ground state. In addition, the vortex state is stable for smaller diameters than the dot resulting in a higher potential bit density for an array of rings. In order for the ring vortex state to be useful, it is necessary to develop a simple method to produce the vortex state and switch the curling direction of this state, which was recently [4] shown to be accomplished by an in-plane magnetic pulse. When considering the response of a ring vortex to a pulse is desirable to have an understanding of the dynamic properties of the vortex state. For this reason, in the following the magnon mode frequencies are calculated on the vortex ground state. This is done using the vortex–magnon interaction including magnetostatic effects as was previously done for the vortex state dot where the core [5] was included. It is remarked that the modes are in the gigahertz range for all modes, and the lower frequency sub gigahertz precessional mode does not appear owing to the absence of the vortex core. We begin by considering small oscillations of the magnetization about the vortex ground state with the magnetostatic field included. For the dot in the vortex ground state the magnetization is expressed ~ ¼ M s ðsin y cos f; sin y sin f; cos yÞ where y as M is the polar angle relative to the ring symmetry axis, and f is the azimuthal angle. These can be written in terms of polar coordinates (r, w) in the dot plane as y ¼ p=2; f ¼ w þ p=2;
(1)
for the vortex ground state. First it is assumed that the magnetization is uniform along the dot axis,
367
then in the continuum approximation the two contributions to the energy can be written as ZZ L ~ 2M ~H ~m d2 x; (2) ½ðA=M 2s ÞðrMÞ W¼ 2 where L is the dot thickness, and the dot radius satisfies the condition, R L: The first term is the contribution from the exchange interaction which is short-range and local in nature. The second term ~m obtained contains the magnetostatic field, H ~m ¼ rF; and it is nonlocal from the potential, H in nature. The sources of this field are both volume ~ and surface charges charges arising from r M ~ at the arising from the normal component of M surface, and it is remarked that for the ground state magnetization given by Eq. (1) there is no magnetostatic charge. Then, for the considerations of excitations for L R the only other contributions to the magnetostatic potential are from the volume and edge contributions, Z ~ ~ rMð r0 Þ 0 0 0 0 Fv ¼ r dr dw dz ; j~ r~ r0 j Z ~ r^MðRÞ dw0 dz0 ; ð3Þ Fe ¼ R j~ r R^rj * r0 Þ2 ¼ r2 þ r0 2 2rr0 cosðw w0 Þ þ where ð r ~ 0 2 ðzz Þ ; and the energy can be expressed as a sum of contributions from exchange, edge, and volume charges, W ¼ W ex þ W e þ W v : Now consider small deviations from the static vortex solutions of the form y ¼ p=2 þ f ðrÞ cosðmw þ otÞ and f ¼ w þ p=2 þ gðrÞ sinðmw þ otÞ
ð4Þ
where the last terms are small, time-dependent corrections. A careful evaluation of the variation, dW =d~ m results in an effective boundary condition and the linearized equation for g. The boundary condition is dg R
LgðRÞ ¼ 0; (5) dr r¼R where L ¼ ðRL=2pl 20 Þ lnð4R=LÞ; and the upper sign is used at the outer edge, Ro and the lower sign is used at the inner edge, Ri : In addition there are two equations for f and g, and after use of the
ARTICLE IN PRESS C.E. Zaspel, B.A. Ivanov / Journal of Magnetism and Magnetic Materials 286 (2005) 366–369
368
approximate expression, f ¼ Og the equation satisfied by g is r2 g þ
O2 1 qFv ; g¼ 2 2 2 qr 4pM s l 0 l0
(6)
where O ¼ o=4pgM s and the exchange length ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi given by l 0 ¼ A 4pM 2s : To orders of approximations considered here, the azimuthal partial derivative in the left-hand side of Eq. (6) is negilgible. For all modes the solution of Eq. (6) without the magnetostatic term are known for any boundary condition is the sum of Bessel functions g0 ðk; rÞ ¼ J m ðkrÞ ¼ sY m ðkrÞ; where k is the wavenumber and s is a constant that will be determined from the boundary conditions at the inner and outer edges. The values of k and s are obtained for the mode m ¼ 1 by numerical solution of the boundary conditions at r ¼ Ri and r ¼ Ro as a function of Ri with Ro ¼ 200 nm: These data are illustrated in Fig. 1 forL ¼ 60 nm: Notice that k is a smoothly increasing function of Ri but the parameter, s is discontinuous owing to the nature of the boundary conditions and Bessel functions. In the following a perturbation technique is developed to obtain the frequency in terms of the nonlocal magnetostatic operators with Fv =4pM s L=R and kl 0 being small parameters. Then from Eq. (6) the following
6
0.06
2 σ
0.04
0
k (1/nm)
4
-2 0.02 -4 20
40
60
80
100 120 140
R i (nm) Fig. 1. The parameter, (solid curves) and the wavenumber (dashed curve) versus the inner radius, Ri for the Ro ¼ 200 nm; L ¼ 60 nm permalloy ring.
expression for the frequency can easily be obtained * + g0 qFv 2 2 ; (7) O g0 ¼ 4pM 2s qr where R R the brackets indicate the integration, h i ¼ 2p 0 r dr: Integration by parts gives the simple expression for the frequency Z Z R Z R 1 R O2 g20 r dr ¼ dr dr0 rðrÞSðr; r0 Þrðr0 Þ; 4p 0 0 0 (8) where rðrÞ ¼ ðd=drÞðrg0 Þ: The function S is obtained from the magnetostatic potential, which after integration over z for the case when LoR is approximately Z 2p cos ma 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi da; Sðr; r Þ ¼ L 2 0 r2 þ r0 þ L2 4 2rr0 cos a (9) which is just a combination of elliptic integrals of modulus k2 ¼ 4rr0 =½ðr þ r0 Þ2 þ L2 =4: At this point it is necessary to use a definite value for m to proceed with a calculation of the frequency, and for m ¼ 1the function S has the form 4L 2 Sðr; r0 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðKðkÞ EðkÞÞ KðkÞ ; k ðr þ r0 Þ2 þ L2 =4
(10) where K and E are elliptic integrals of the first and second kind. First consider m ¼ 1 mode, which has a symmetry that allows excitation by an in-plane magnetic pulse. Using the effective boundary condition for m ¼ 1; the values of k and s from Fig. 1 are used in the numerical integration of Eq. (8) to obtain the frequency. When this is done one obtains the frequency as a function of Ri for the ring of outer radius, Ro ¼ 200 nm and of thickness, L ¼ 60 nm: Referring to these results in Fig. 2, notice that the frequency is a slowly increasing function of the inner radius up to about Ri ¼ 140 nm: At this radius the wavenumber has increased so that kl 0 is no longer a small parameter and the approximations used for the derivation of Eq. (8) break down. Another interesting mode is m ¼ 0 mode that can be excited by thermal
ARTICLE IN PRESS C.E. Zaspel, B.A. Ivanov / Journal of Magnetism and Magnetic Materials 286 (2005) 366–369
the curling direction in the vortex-state ring. For this reason, m ¼ 1 mode can possibly have effects on the dynamics of vortex switching. Other modes not considered here are the higher frequency m ¼ 1 modes with additional radial nodes as well as the higher m modes. However, from earlier observations of dot modes [3] it is remarked that these will probably not be observed owing to their low amplitude.
m=0
Freq. (GHz)
10 m=1
8
369
6 4
40
80 Ri (nm)
120
160
Fig. 2. Frequencies of m ¼ 0 and 1 modes versus the inner radius.
fluctuations or an external pulse perpendicular to the ring plane. The frequency of this mode is calculated in the same way and the results are also illustrated in Fig. 2. In conclusion, m ¼ 1 mode is of the lowest frequency; moreover, this mode is also excited by an in-plane pulse which has been shown to reverse
This work was supported by National Science Foundation Grants numbers DMR-9972507 and DMR-9974273. References [1] W.J. Gallagher, S.S.P. Parkin, Y. Lu, X.P. Bian, A. Marley, K.P. Roche, R.A. Altman, S.A. Rishton, C. Jahnes, T.M. Shaw, G. Xiao, J. Appl. Phys. 81 (1997) 3741. [2] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, T. Ono, Science 289 (2000) 930. [3] J.P. Park, P. Eames, D.M. Engebretson, J. Berezovsky, P.A. Crowell, Phys. Rev. B 67 (2003) 020403(R). [4] U. Welp, V.K. Vlasko-Vlasov, J.M. Miller, N.J. Zaluzec, V. Metlushko, B. Ilic, Phys. Rev. B 68 (2003) 054408. [5] B.A. Ivanov, C.E. Zaspel, Appl. Phys. Lett. 81 (2002) 1261.
Journal of Magnetism and Magnetic Materials 286 (2005) 366–369 www.elsevier.com/locate/jmmm
Magnon modes in permalloy nanorings C.E. Zaspela,, B.A. Ivanovb a
Department of Physics, Montana State University, Bozeman, MT 59717, USA b Institute of Magnetism, NASU, 36B Vernadskii Ave. 03142 Kiev, Ukraine Available online 4 November 2004
Abstract Submicron permalloy rings can have a vortex ground state as a result of the competing effects arising from the exchange and magnetostatic interactions. The dynamic properties of this state are investigated by analytical calculations of the structure and frequencies of the magnon normal modes in the presence of the vortex–magnon interaction. For thin rings of thickness, L ¼ 60 nm and inner and outer radii, Ri and Ro=200 nm, respectively it is shown that the magnon node frequencies in the 10 GHz range for rings of radii of a few hundred nanometers. Moreover, the frequencies are slowly increasing functions of Ro for fixed Ro. r 2004 Published by Elsevier B.V. PACS: 75.30.Ds; 75.40.Gb; 75.75.+a Keywords: Spin waves; Magnetic vortex; Permalloy rings
Magnetic microdots made from a soft ferromagnetic material such as permalloy have been proposed [1] for use as magnetic storage media, and the study of the basic physics of magnetic particles in this size range has led to interesting effects. For example, when the dot radius is reduced to the micron and submicron range the competition between exchange and the magnetostatic interaction will result in a vortex ground state with the magnetization mainly confined to the dot plane. This structure is stable because the Corresponding author. Tel: 406 994 3614; fax: 406 994 4452.
E-mail address: [email protected] (C.E. Zaspel). 0304-8853/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.jmmm.2004.09.093
curling effect in the planar vortex will eliminate effective magnetostatic charges thereby minimizing the energy. However, at the core of the planar vortex there is a ‘energy singularity, that is eliminated by an out of plane rotation of the magnetization at the core resulting in a small surface charge, which has also been observed [2] in thin dots. In materials such as permalloy the presence of the core results in a net magnetic moment perpendicular to the dot plane over a central region of about 10 nm diameter. If arrays of dots are to be used in magnetic information storage, this vortex core presents several disadvantages. The net magnetic moment of the dot will
ARTICLE IN PRESS C.E. Zaspel, B.A. Ivanov / Journal of Magnetism and Magnetic Materials 286 (2005) 366–369
interact through the dipolar interaction with other dots in the array affecting properties such as switching of the curling direction. Because of the presence of the core, there is a low-frequency (sub gigahertz) precessional mode [3] that will complicate the dynamics of the single dot as well as the array of dots. Finally, the vortex state is only stable above a critical diameter of approximately 100 nm limiting the possible bit density in arrays of these systems. Some of these problems can be overcome by the use of a ring rather than a dot. The ring has no core, and therefore no magnetostatic fields other than those from fluctuations about the vortex ground state. In addition, the vortex state is stable for smaller diameters than the dot resulting in a higher potential bit density for an array of rings. In order for the ring vortex state to be useful, it is necessary to develop a simple method to produce the vortex state and switch the curling direction of this state, which was recently [4] shown to be accomplished by an in-plane magnetic pulse. When considering the response of a ring vortex to a pulse is desirable to have an understanding of the dynamic properties of the vortex state. For this reason, in the following the magnon mode frequencies are calculated on the vortex ground state. This is done using the vortex–magnon interaction including magnetostatic effects as was previously done for the vortex state dot where the core [5] was included. It is remarked that the modes are in the gigahertz range for all modes, and the lower frequency sub gigahertz precessional mode does not appear owing to the absence of the vortex core. We begin by considering small oscillations of the magnetization about the vortex ground state with the magnetostatic field included. For the dot in the vortex ground state the magnetization is expressed ~ ¼ M s ðsin y cos f; sin y sin f; cos yÞ where y as M is the polar angle relative to the ring symmetry axis, and f is the azimuthal angle. These can be written in terms of polar coordinates (r, w) in the dot plane as y ¼ p=2; f ¼ w þ p=2;
(1)
for the vortex ground state. First it is assumed that the magnetization is uniform along the dot axis,
367
then in the continuum approximation the two contributions to the energy can be written as ZZ L ~ 2M ~H ~m d2 x; (2) ½ðA=M 2s ÞðrMÞ W¼ 2 where L is the dot thickness, and the dot radius satisfies the condition, R L: The first term is the contribution from the exchange interaction which is short-range and local in nature. The second term ~m obtained contains the magnetostatic field, H ~m ¼ rF; and it is nonlocal from the potential, H in nature. The sources of this field are both volume ~ and surface charges charges arising from r M ~ at the arising from the normal component of M surface, and it is remarked that for the ground state magnetization given by Eq. (1) there is no magnetostatic charge. Then, for the considerations of excitations for L R the only other contributions to the magnetostatic potential are from the volume and edge contributions, Z ~ ~ rMð r0 Þ 0 0 0 0 Fv ¼ r dr dw dz ; j~ r~ r0 j Z ~ r^MðRÞ dw0 dz0 ; ð3Þ Fe ¼ R j~ r R^rj * r0 Þ2 ¼ r2 þ r0 2 2rr0 cosðw w0 Þ þ where ð r ~ 0 2 ðzz Þ ; and the energy can be expressed as a sum of contributions from exchange, edge, and volume charges, W ¼ W ex þ W e þ W v : Now consider small deviations from the static vortex solutions of the form y ¼ p=2 þ f ðrÞ cosðmw þ otÞ and f ¼ w þ p=2 þ gðrÞ sinðmw þ otÞ
ð4Þ
where the last terms are small, time-dependent corrections. A careful evaluation of the variation, dW =d~ m results in an effective boundary condition and the linearized equation for g. The boundary condition is dg R
LgðRÞ ¼ 0; (5) dr r¼R where L ¼ ðRL=2pl 20 Þ lnð4R=LÞ; and the upper sign is used at the outer edge, Ro and the lower sign is used at the inner edge, Ri : In addition there are two equations for f and g, and after use of the
ARTICLE IN PRESS C.E. Zaspel, B.A. Ivanov / Journal of Magnetism and Magnetic Materials 286 (2005) 366–369
368
approximate expression, f ¼ Og the equation satisfied by g is r2 g þ
O2 1 qFv ; g¼ 2 2 2 qr 4pM s l 0 l0
(6)
where O ¼ o=4pgM s and the exchange length ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi given by l 0 ¼ A 4pM 2s : To orders of approximations considered here, the azimuthal partial derivative in the left-hand side of Eq. (6) is negilgible. For all modes the solution of Eq. (6) without the magnetostatic term are known for any boundary condition is the sum of Bessel functions g0 ðk; rÞ ¼ J m ðkrÞ ¼ sY m ðkrÞ; where k is the wavenumber and s is a constant that will be determined from the boundary conditions at the inner and outer edges. The values of k and s are obtained for the mode m ¼ 1 by numerical solution of the boundary conditions at r ¼ Ri and r ¼ Ro as a function of Ri with Ro ¼ 200 nm: These data are illustrated in Fig. 1 forL ¼ 60 nm: Notice that k is a smoothly increasing function of Ri but the parameter, s is discontinuous owing to the nature of the boundary conditions and Bessel functions. In the following a perturbation technique is developed to obtain the frequency in terms of the nonlocal magnetostatic operators with Fv =4pM s L=R and kl 0 being small parameters. Then from Eq. (6) the following
6
0.06
2 σ
0.04
0
k (1/nm)
4
-2 0.02 -4 20
40
60
80
100 120 140
R i (nm) Fig. 1. The parameter, (solid curves) and the wavenumber (dashed curve) versus the inner radius, Ri for the Ro ¼ 200 nm; L ¼ 60 nm permalloy ring.
expression for the frequency can easily be obtained * + g0 qFv 2 2 ; (7) O g0 ¼ 4pM 2s qr where R R the brackets indicate the integration, h i ¼ 2p 0 r dr: Integration by parts gives the simple expression for the frequency Z Z R Z R 1 R O2 g20 r dr ¼ dr dr0 rðrÞSðr; r0 Þrðr0 Þ; 4p 0 0 0 (8) where rðrÞ ¼ ðd=drÞðrg0 Þ: The function S is obtained from the magnetostatic potential, which after integration over z for the case when LoR is approximately Z 2p cos ma 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi da; Sðr; r Þ ¼ L 2 0 r2 þ r0 þ L2 4 2rr0 cos a (9) which is just a combination of elliptic integrals of modulus k2 ¼ 4rr0 =½ðr þ r0 Þ2 þ L2 =4: At this point it is necessary to use a definite value for m to proceed with a calculation of the frequency, and for m ¼ 1the function S has the form 4L 2 Sðr; r0 Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðKðkÞ EðkÞÞ KðkÞ ; k ðr þ r0 Þ2 þ L2 =4
(10) where K and E are elliptic integrals of the first and second kind. First consider m ¼ 1 mode, which has a symmetry that allows excitation by an in-plane magnetic pulse. Using the effective boundary condition for m ¼ 1; the values of k and s from Fig. 1 are used in the numerical integration of Eq. (8) to obtain the frequency. When this is done one obtains the frequency as a function of Ri for the ring of outer radius, Ro ¼ 200 nm and of thickness, L ¼ 60 nm: Referring to these results in Fig. 2, notice that the frequency is a slowly increasing function of the inner radius up to about Ri ¼ 140 nm: At this radius the wavenumber has increased so that kl 0 is no longer a small parameter and the approximations used for the derivation of Eq. (8) break down. Another interesting mode is m ¼ 0 mode that can be excited by thermal
ARTICLE IN PRESS C.E. Zaspel, B.A. Ivanov / Journal of Magnetism and Magnetic Materials 286 (2005) 366–369
the curling direction in the vortex-state ring. For this reason, m ¼ 1 mode can possibly have effects on the dynamics of vortex switching. Other modes not considered here are the higher frequency m ¼ 1 modes with additional radial nodes as well as the higher m modes. However, from earlier observations of dot modes [3] it is remarked that these will probably not be observed owing to their low amplitude.
m=0
Freq. (GHz)
10 m=1
8
369
6 4
40
80 Ri (nm)
120
160
Fig. 2. Frequencies of m ¼ 0 and 1 modes versus the inner radius.
fluctuations or an external pulse perpendicular to the ring plane. The frequency of this mode is calculated in the same way and the results are also illustrated in Fig. 2. In conclusion, m ¼ 1 mode is of the lowest frequency; moreover, this mode is also excited by an in-plane pulse which has been shown to reverse
This work was supported by National Science Foundation Grants numbers DMR-9972507 and DMR-9974273. References [1] W.J. Gallagher, S.S.P. Parkin, Y. Lu, X.P. Bian, A. Marley, K.P. Roche, R.A. Altman, S.A. Rishton, C. Jahnes, T.M. Shaw, G. Xiao, J. Appl. Phys. 81 (1997) 3741. [2] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, T. Ono, Science 289 (2000) 930. [3] J.P. Park, P. Eames, D.M. Engebretson, J. Berezovsky, P.A. Crowell, Phys. Rev. B 67 (2003) 020403(R). [4] U. Welp, V.K. Vlasko-Vlasov, J.M. Miller, N.J. Zaluzec, V. Metlushko, B. Ilic, Phys. Rev. B 68 (2003) 054408. [5] B.A. Ivanov, C.E. Zaspel, Appl. Phys. Lett. 81 (2002) 1261.