- Email: [email protected]
Micromagnetics and interaction effects in the lattice of magnetic dots
Journal of Magnetism and Magnetic Materials 215}216 (2000) 37}39
Micromagnetics and interaction e!ects in the lattice of magnetic dots Konstantin L. Metlov* Institute of Physics ASCR, Na Slovance 2, 18040 Prague 8, Czech Republic
Abstract This work reports micromagnetic calculation of the distribution of stray "elds in a regular planar array of cylindrical magnetic dots uniformly magnetized in the plane of the array. Explicit analytical expressions for distribution of stray "elds are given. 2000 Elsevier Science B.V. All rights reserved. Keywords: Stray "elds; Magnetic dots
Recently, the microscopic structures made of magnetic dots attained interest due to their possible applications for random access magnetic memory devices. For constructing such devices and especially for estimating their recording density limits, it is necessary to study the e!ects of inter-dot magnetic coupling, which in#uence the density of recording. This paper reports micromagnetic calculations of the internal stray "elds in the periodic regular array of uniformly magnetized magnetic dots, for the case of magnetization vector lying in the array plane. The calculation is performed analytically and takes into account all neighbors of any dot and all details of the stray "eld of a single dot. The method of this calculation is similar to the one used in calculations of periodic domain structures in magnetic garnets [1]. First, consider the geometry of the task. A schematic illustration of the dots array is shown in Fig. 1. The origin of the coordinate system is chosen in the center of a dot, the X> plane coincides with the array plane (the X-axis is horizontal in the "gure), the Z-axis is perpendicular to the array plane. The dots are circular cylinders of radius R and thickness h, arranged into a square lattice with the period D. It is assumed that all dots are uniformly magnetized, moreover, the magnetization of all dots is the same and lying in the array plane. The angle between the magnetization vector and the axis X is denoted as h. * Tel.: #420-2-66052617; fax: #420-2-821227. E-mail address: [email protected] (K.L. Metlov).
We consider the stationary distribution of magnetization and there are no macroscopical currents in the system. This allows us to use the magnetic charge formalism and introduce the scalar potential for the magnetic "eld ;(r). The magnetic "eld itself can then be expressed as H"e;(r), where e"+R , R , R , is a gradient operV W X ator. Thus, to "nd the distribution of the stray "eld we need to "nd the corresponding scalar magnetic potential ;(r). It is known from the textbooks (see e.g. Ref. [2]), that the scalar magnetic potential satis"es the Poisson equation e;(r)"c e ) M(r), where c "4p for CGS system of units (c "1 in SI). This equation should be solved with the following boundary conditions: "r";(r)(R and r"e;(r)"(R when "r"PR along any path. On the boundaries of magnetic material the scalar magnetic potential should be a continuous function of coordinate and its normal (to the surface of material) derivative R;/Rn should have a jump, proportional to the normal component of the magnetization vector inside the material, c M ) n. The distribution of volume magnetic charge o(r, )"c e ) M(r) of one dot (inside the unit cell of the array) can be written as o(r, h),o(x, y, z, h),o(r, , z, h) "!c
r M 1 d 1! R R
2z 2z ;cos( !h)H 1! H 1# , h h
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 0 6 0 - 3
(1)
38
K.L. Metlov / Journal of Magnetism and Magnetic Materials 215}216 (2000) 37}39
image of ;. D o KL ; (z)" KL 16p m#n
;
Fig. 1. The illustration of the dots array showing parameters used in this calculation. Distributions of magnetic charges and the force lines of the "eld by a single dot are shown in the center of this "gure.
o (k ) D KL 8 . ; (k )" KL 8 (2p) m#n#(k D) 8
(2)
Calculating the Fourier integrals over z and k one 8 may obtain the following expression for the Fourier
2p"z" ph exp ! (m#n sinh (m#n , "z"'h/2. D D
(3)
It is easy to check that both ; (z) and its derivative with KL respect to z are continuous at the surfaces of the "lm (z"$h/2). It is because in the case of in-plane magnetization there are no magnetic charges at these surfaces. Let us now evaluate o . According to the de"nition of KL Fourier coe$cients we have
4 " " o " dx dy o(x, y) KL D \" \" ;sin
where M is the saturation magnetization of the material, 1 H(x) is the generalized unit step function (H(x'0)"1, H(x(0)"0), d(x)"dH(x)/dx, the other parameters are shown in Fig. 1. Note that all charge is concentrated on the boundary of the dot and that we used cylindrical coordinates with the origin at the dot center. Because the Poisson equation is linear and the charges at any arbitrary angle of magnetization vector h can be represented as o(x, y, z, h)"cos(h)o(x, y, z, 0)# sin(h)o(y, x, z, 0) the scalar magnetic potential (and the distribution of stray "elds) also has a similar angular dependence. Thus, all following consideration will be done for the case h"0. To elaborate the periodicity of the array we shall represent ; and o by the Fourier series in the X> plane, and by the Fourier integral in the Z direction, thus introducing their Fourier transforms ; (k ), o (k ). KL 8 KL 8 Moreover, both o and ; are antisymmetric in the X direction and symmetric in both the > and Z directions (recall that we now consider the case when h"0). This means that we shall use sin Fourier transform in the X direction and cos in the > and Z directions, de"ning the ; and o in a way that they are real KL KL numbers. By inserting the Fourier representations of ; and o the Poisson equation for scalar magnetic potential can be rewritten as
ph pz 1!exp ! (m#n cosh 2 (m#n , "z" )h/2 D D
2px 2py m cos n D D
8pc M R m 1 " J D (m#n
;
2pR (m#n . D
(4)
Fig. 2. Contour plot of the scalar magnetic potential inside the dots array, at z"0, h/D"0.045, R/D"0.45. Values of the potential are normalized to c M D. Vertical lines (e.g. at 1 x"0,$0.5,2) correspond to zero value of the potential, step between contours is 0.01, values of potential in the left half of each dot are negative while those in the right half are positive.
K.L. Metlov / Journal of Magnetism and Magnetic Materials 215}216 (2000) 37}39
Here the formula (3.711) in Ref. [3] was used and J denotes the Bessel function of the "rst kind and the "rst order. Finally, the resulting scalar magnetic potential in coordinate space can be written as &back' Fourier transform of ; (z) KL 2py ;(r),;(x, y, z)" ; (z) cos n KL D K L ; (z) 2px # K sin m . (5) 2 D
At a limit DPR this expression coincides with an expression for a scalar magnetic potential of an isolated dot magnetized in-plane, which, in turn, asymptotically at zPR produces "elds equivalent to a dipole with a moment pM hR (CGS units). Note that even for 1 a single uniformly magnetized dot the stray "eld at "nite distance will contain higher multipole terms. The example of resulting distribution of the scalar magnetic potential within the array is shown in Fig. 2. The reported expressions take into account the in#uence of all neighbors of any dot and all details of the stray
39
"eld of a single dot. They may be used as a checking point for numerical micromagnetic simulations and also for futher evaluation of the in#uence of inter-dot coupling on spin-wave modes of the array of cylindrical magnetic dots. Similarly, the stray "eld in other periodic magnetic structures of magnetic dots of di!erent shapes can be considered. This work was supported in part by the Grant Agency of the Czech Republic under project 202/99/P052 and the joint project No. ME..140(1998) by MSMT CR and NSF USA.
References [1] W.F. Druyvesteyn, J.W.F. Dorleijn, Philips Res. Rep. 26 (1971) 11. [2] A. Aharoni, Introduction to the theory of ferromagnetism, Oxford University Press, Oxford, 1996, ISBN 0 19 851791 2. [3] I.S. Gradshtejn, I.M. Ryzhik, Tables of series, products, and integrals, Gos. izdatelstvo "z.-mat. literatury, Moskva, 1963.
Micromagnetics and interaction e!ects in the lattice of magnetic dots Konstantin L. Metlov* Institute of Physics ASCR, Na Slovance 2, 18040 Prague 8, Czech Republic
Abstract This work reports micromagnetic calculation of the distribution of stray "elds in a regular planar array of cylindrical magnetic dots uniformly magnetized in the plane of the array. Explicit analytical expressions for distribution of stray "elds are given. 2000 Elsevier Science B.V. All rights reserved. Keywords: Stray "elds; Magnetic dots
Recently, the microscopic structures made of magnetic dots attained interest due to their possible applications for random access magnetic memory devices. For constructing such devices and especially for estimating their recording density limits, it is necessary to study the e!ects of inter-dot magnetic coupling, which in#uence the density of recording. This paper reports micromagnetic calculations of the internal stray "elds in the periodic regular array of uniformly magnetized magnetic dots, for the case of magnetization vector lying in the array plane. The calculation is performed analytically and takes into account all neighbors of any dot and all details of the stray "eld of a single dot. The method of this calculation is similar to the one used in calculations of periodic domain structures in magnetic garnets [1]. First, consider the geometry of the task. A schematic illustration of the dots array is shown in Fig. 1. The origin of the coordinate system is chosen in the center of a dot, the X> plane coincides with the array plane (the X-axis is horizontal in the "gure), the Z-axis is perpendicular to the array plane. The dots are circular cylinders of radius R and thickness h, arranged into a square lattice with the period D. It is assumed that all dots are uniformly magnetized, moreover, the magnetization of all dots is the same and lying in the array plane. The angle between the magnetization vector and the axis X is denoted as h. * Tel.: #420-2-66052617; fax: #420-2-821227. E-mail address: [email protected] (K.L. Metlov).
We consider the stationary distribution of magnetization and there are no macroscopical currents in the system. This allows us to use the magnetic charge formalism and introduce the scalar potential for the magnetic "eld ;(r). The magnetic "eld itself can then be expressed as H"e;(r), where e"+R , R , R , is a gradient operV W X ator. Thus, to "nd the distribution of the stray "eld we need to "nd the corresponding scalar magnetic potential ;(r). It is known from the textbooks (see e.g. Ref. [2]), that the scalar magnetic potential satis"es the Poisson equation e;(r)"c e ) M(r), where c "4p for CGS system of units (c "1 in SI). This equation should be solved with the following boundary conditions: "r";(r)(R and r"e;(r)"(R when "r"PR along any path. On the boundaries of magnetic material the scalar magnetic potential should be a continuous function of coordinate and its normal (to the surface of material) derivative R;/Rn should have a jump, proportional to the normal component of the magnetization vector inside the material, c M ) n. The distribution of volume magnetic charge o(r, )"c e ) M(r) of one dot (inside the unit cell of the array) can be written as o(r, h),o(x, y, z, h),o(r, , z, h) "!c
r M 1 d 1! R R
2z 2z ;cos( !h)H 1! H 1# , h h
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 0 6 0 - 3
(1)
38
K.L. Metlov / Journal of Magnetism and Magnetic Materials 215}216 (2000) 37}39
image of ;. D o KL ; (z)" KL 16p m#n
;
Fig. 1. The illustration of the dots array showing parameters used in this calculation. Distributions of magnetic charges and the force lines of the "eld by a single dot are shown in the center of this "gure.
o (k ) D KL 8 . ; (k )" KL 8 (2p) m#n#(k D) 8
(2)
Calculating the Fourier integrals over z and k one 8 may obtain the following expression for the Fourier
2p"z" ph exp ! (m#n sinh (m#n , "z"'h/2. D D
(3)
It is easy to check that both ; (z) and its derivative with KL respect to z are continuous at the surfaces of the "lm (z"$h/2). It is because in the case of in-plane magnetization there are no magnetic charges at these surfaces. Let us now evaluate o . According to the de"nition of KL Fourier coe$cients we have
4 " " o " dx dy o(x, y) KL D \" \" ;sin
where M is the saturation magnetization of the material, 1 H(x) is the generalized unit step function (H(x'0)"1, H(x(0)"0), d(x)"dH(x)/dx, the other parameters are shown in Fig. 1. Note that all charge is concentrated on the boundary of the dot and that we used cylindrical coordinates with the origin at the dot center. Because the Poisson equation is linear and the charges at any arbitrary angle of magnetization vector h can be represented as o(x, y, z, h)"cos(h)o(x, y, z, 0)# sin(h)o(y, x, z, 0) the scalar magnetic potential (and the distribution of stray "elds) also has a similar angular dependence. Thus, all following consideration will be done for the case h"0. To elaborate the periodicity of the array we shall represent ; and o by the Fourier series in the X> plane, and by the Fourier integral in the Z direction, thus introducing their Fourier transforms ; (k ), o (k ). KL 8 KL 8 Moreover, both o and ; are antisymmetric in the X direction and symmetric in both the > and Z directions (recall that we now consider the case when h"0). This means that we shall use sin Fourier transform in the X direction and cos in the > and Z directions, de"ning the ; and o in a way that they are real KL KL numbers. By inserting the Fourier representations of ; and o the Poisson equation for scalar magnetic potential can be rewritten as
ph pz 1!exp ! (m#n cosh 2 (m#n , "z" )h/2 D D
2px 2py m cos n D D
8pc M R m 1 " J D (m#n
;
2pR (m#n . D
(4)
Fig. 2. Contour plot of the scalar magnetic potential inside the dots array, at z"0, h/D"0.045, R/D"0.45. Values of the potential are normalized to c M D. Vertical lines (e.g. at 1 x"0,$0.5,2) correspond to zero value of the potential, step between contours is 0.01, values of potential in the left half of each dot are negative while those in the right half are positive.
K.L. Metlov / Journal of Magnetism and Magnetic Materials 215}216 (2000) 37}39
Here the formula (3.711) in Ref. [3] was used and J denotes the Bessel function of the "rst kind and the "rst order. Finally, the resulting scalar magnetic potential in coordinate space can be written as &back' Fourier transform of ; (z) KL 2py ;(r),;(x, y, z)" ; (z) cos n KL D K L ; (z) 2px # K sin m . (5) 2 D
At a limit DPR this expression coincides with an expression for a scalar magnetic potential of an isolated dot magnetized in-plane, which, in turn, asymptotically at zPR produces "elds equivalent to a dipole with a moment pM hR (CGS units). Note that even for 1 a single uniformly magnetized dot the stray "eld at "nite distance will contain higher multipole terms. The example of resulting distribution of the scalar magnetic potential within the array is shown in Fig. 2. The reported expressions take into account the in#uence of all neighbors of any dot and all details of the stray
39
"eld of a single dot. They may be used as a checking point for numerical micromagnetic simulations and also for futher evaluation of the in#uence of inter-dot coupling on spin-wave modes of the array of cylindrical magnetic dots. Similarly, the stray "eld in other periodic magnetic structures of magnetic dots of di!erent shapes can be considered. This work was supported in part by the Grant Agency of the Czech Republic under project 202/99/P052 and the joint project No. ME..140(1998) by MSMT CR and NSF USA.
References [1] W.F. Druyvesteyn, J.W.F. Dorleijn, Philips Res. Rep. 26 (1971) 11. [2] A. Aharoni, Introduction to the theory of ferromagnetism, Oxford University Press, Oxford, 1996, ISBN 0 19 851791 2. [3] I.S. Gradshtejn, I.M. Ryzhik, Tables of series, products, and integrals, Gos. izdatelstvo "z.-mat. literatury, Moskva, 1963.