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Magnetic bubble domain interactions
Solid State Communications,
Vol. 8, PP. 1303—1306, 1970.
Pergamon Press.
Printed in Great Britain
MAGNETIC BUBBLE DOMAIN INTERACTIONS J.A. Cape and G.W. Lehman Science Center, North American Rockwell Corporation, Thousand Oaks, California 91360
(Received 1 June 1970 by A.A. Maradudin)
We consider the interaction between cylindrical magnetic (“bubble”) domains and predict a stable triangular lattice configuration in an applied field H~ 1 < H0 < H~2. For 0 < H~ < ~ a stripe pattern is stable. Detailed predictions agree very well with experiment.
RECENTLY much attention has been drawn to the formation and manipulation of cylindrical domains in the rare earth orthoferrites.’ These “bubble” domains are also foundininthin other 25 generally occuring platelets materials, with a perpendicular easy axis and in the presence of a (relatively high) external magnetic field also normal to the plate. The stability of isolated (non-interfacing) bubble domains was
characteristic lengths structure in the walls, domain configuartion: sotropy stant, constant, and
first3 investigated Kooy and recentlytheoretically extended in by detail by and Thiele.6 Enzconsideration was given to the interaction No
can be written as: ‘7
governing respectively the and the macroscopic K,.. is the uniaxial aniA0 is the exchange con-
The total magnetostatic energy in this model of a single reverse domain of volume v and wall area A in the presence of an external field H
0 u
compare our results with existing experimental data, The phenomenological model we employ to3’68 treatconsider the domains has appliedh elsewhere. We a plate of been thickness and infinite lateral dimensions and assign a surface energy a~ to the domain walls which are assumed to be everywhere perpendicular to the surface. Implicit in the model is the assumption that 2irM.~ =
K
‘-~sat
A
1 2 = — v[1 — N — H~]+ —~- (2) 87TM8 Here ~ is the energy of the plate with no reverse domains, N is the ~average)demagnetization factor of the domain, and fi 0 H0/4~M~. For cylindrical domains the factor N depends only on the ratio of the diameter d to the height h. Thus u1 is to. be minimized with respect to d/h for to determine of which H0 for the given ~/h which u the < 0;values i.e., for domain is stable.”3’6 When u 1 < 0, it is clear that a multiplicity of domains is favored with the equilibrium domain density limited by the interdomain interaction.
between bubbles which the might be and expected to influence substantially size distribution of the domains. In the following we discuss some of our theoretical studies of the interaction and
~
—
Of the possible “monatomic” space filling =
<< 1
(1)
lattices of cylinders, it can be shown that the lowest energy configuration is the triangular (closest that casepacking)~lattice. can be written as:The energy density in
Here M3 is the spontaneous 1 /2, and ~ = magnetization, are the = 2(A~/K,~)
1303
1304
MAGNETIC BUBBLE DOMAIN INTERACTIONS ULatticeU~at 2V 8nM~ —
(_~__)(i ‘~‘ceU
H0
—
____
m,n
2E~-)+(.~x_)2
____
q
(q~~y2)2
an increasing field as H 0 exceeds H~1,or, in a decreasing field as H 0 becomes less than He,. Rather, various artificial 1 ,5 means have been
=
J~(q~y/2) (1- e~x) x
Vol.8, No. 16
used to produce an abundance of bubbles, (e.g., the “cutting” of stripes with a magnetized needle). (3)
The fact that the bubble lattice does not
mn
where x = h/a, y = d/a, a is the 2/2 nearest is the unit neighbor distance, V~ = \/(3) a cell volume, and q~ = 477/~/(3) (m2 + mn + 02)’ 2• The energy tunction, equation (3) is to be minimized with respect to x and y to determine the stationary values of d and a for given H0 and ~ /h. Some of our results are shown on Fig. 1 and other results and conclusions are summarized qualitatively in the following. At H0 = 0, the cylindrical domain lattice is energetically less favorable 3(byThis roughly 1—5%) situation than the familiar stripe pattern. obtains for all H 0 ~ H~,(region I). At H~ = H~ (at roughly 1/3 the saturation field) the bubble lattice energy density equals the stripe energy density, and the bubble lattice is then the favored structure throughout region II, H~,~ H0 < H~2,where H~2 is defined by
~ H
~‘Lottice 11c2
—
Usat
=
u,
=
0. In region III,
0 < H~3,isolated bubbles may exist metastably; i.e., an energy barrier prevents their collapse. At H~ H~2 and h/e 4 (where the domain diameters are smallest; see Fig. 1) this2M~h3. barrierFor height is about in units of example, for 0.1 TmFeO 2ir 3 at h/~ = 4, the barrier is about 6 x 10~ergs/bubble. It should be noted that the same barrier inhibits the nucleation of bubble domains. Thus, one may expect that in the applied filed range from H~3 to somewhat less than H02 ,~ a sparse density of weakly interacting bubbles can be maintained while individual bubbles may be artificially created and destroyed, fixed in position, or moved about by external means with the expectation that none will appear or disappear
appear spontaneously when it is the lower energy structure and the fact that it has been observed at H 0 = 0 where it is not the lowest energy structure are complications related to the mechanism of the nucleation, the existence of wallcoercivity in non-ideal materials, and the fact that conversion of stripe arrays to bubble arrays involves the formation of (temporary), excess walls and hence an intermediate energy barrier. Discussion of these matters is beyond the scope of this letter. Our main purpose is to point out that the present calculations agree well with published data lattices in those have casesbeen where essentially regular bubble observed and the numerical values reported. Given the plate thickness, external magnetic field, and the values of 47r&15 and ~, the present model is used to predict the nearest neighbor distance a and domain diameter d which may be compared with observation. We have been able to find only three cases for which sufficient data were presented. Interestingly, the data are for three basically different magnetic materials. The results are shown in Table 1. The good agreement may be suprising in view of the methods resorted to, to produce the bubbles in these experiments. In this regard, our calculations show that over a wide range in H0, the bubble lattice “fits” the iso-field stripe array, i.e., the distance between stripes equals the distance between bubble rows. This situation is undoubtedly conductive to the production of a nearly equilibrium bubble array. A detailed presentation of our calculations 1
10
spontaneously.’
will be published eisewhere.
In practice, it is typically observed that the bubble lattice does not appear spontaneously in
Acknowledgements — we are indebted to L. Vredevoe for valuable discussion.
Vol.8, No.16
MAGNETIC BUBBLE DOMAIN INTERACTIONS I
~1~
I
I I
40
~
1305 I -
~m,
________
h/~
FIG. 1. The reduced critical fields as a function of the reduced plate thickness, and the corresponding critical bubble diameters. The fields are defined by: H 01, the bubble lattice energy density equals the stripe domain energy density; H02 , the bubble energy equals zero; and H03 , the energy barrier stabilizing isolated bubbles vanishes. The curve d0 is the metastable bubble diameter at H0 = 0. Curve d03 has been attributed to Thiele (see reference 1).
ic1ble 1. Comparison of observed and calculated values of bubble diameter and separation. Material BaFe12019 TmFeO 3 GdIG
*
Ref. 3 m 1 5,7
d(obs.)
*
4cm
(0.5— x lO 3x 101.0) ~ l. 4.5 x 103cm
a(obs.)* (7.0) x lO4cm
d(calc.) 1 x 104cm
a(calc.) 6.7 x lO4cmt
4.3 x 10~cm
1.26 x 102cm~
4.3 x 10~cm~
6.0 x 103cm
4.5 x 103cm §
6.1 x 103cm~
Average values in some cases. Includes correction for finite geometry of plate but none for (27TM. 2/K~) 0 (see reference 3). 9 Applied field not given in reference 1. Calculated results are best fit at U = 0.245.
§
Present calculated values based on
2.5 x i0~, roughly twice the value given in reference 7
=
which is thought to be in error due to computational errors in that work; see reference 7. REFERENCES 1.
BOBECK A.H., Bell Sys. Tech. J. 46, 1901 (1967); BOBEK A.H., FISCHER R.F., PERNESKI A.J., REMEIKA J.P. and Van UITERT L.G., IEEE Transactions on Magnetics, Mag. 5, 544 (1969).
2.
SHERWOOD R.C., REMEIKA J.P. and WILLIAMS H.J., J. appi. Phys. 30, 217 (1959).
3.
KOOY C. and ENZ U., Philips Res. Rep. 15, 7 (1960).
4.
MEE C.D., Contemp. Phys. 8, 385 (1967).
1306
MAGNETIC BUBBLE DOMAIN INTERACTIONS
Vol.8, No.16
5.
NEMCHIK J.M., J. appi. Phys. 40, 1086 (1969).
6.
THIELE., Bell Sys. Tech. J. 48, 3287 (1969).
7.
CHARAP S.H. and NEMCHIK J.M., IEEE Transactions on Magnetics, Mag. 5, 566 (1969). These authors have also derived equation (3) for the energy. However, we find that their numerical solution of the minimization of equation (3) is in error. A detailed discussion of the computational problem and the source of errors will be discussed in a full article (reference 10).
8.
For a review of the theory of ferromagnetic domains, see KITTEL C., Rev, mod. Phys. 21, 541 (1949).
9.
The exact lower field limit of stability of an isolated bubble has been determined by THIELE, reference 6.
10.
CAPE J.A. and LEHMAN G.W., manuscript in preparation.
Nous avons consideré l’interaction entre des domaines magnitiques de geometrie cylindrique (“Bubbles”) et prédit une configuration de reseau triangulaire stable dans une champ appliqué, H 01 < H0 ‘102. Pour 0 < H0 < H01 un reseau de bandes est stable. Les predictions detaillés son en bon accord avec les resultats d’experience.
Vol. 8, PP. 1303—1306, 1970.
Pergamon Press.
Printed in Great Britain
MAGNETIC BUBBLE DOMAIN INTERACTIONS J.A. Cape and G.W. Lehman Science Center, North American Rockwell Corporation, Thousand Oaks, California 91360
(Received 1 June 1970 by A.A. Maradudin)
We consider the interaction between cylindrical magnetic (“bubble”) domains and predict a stable triangular lattice configuration in an applied field H~ 1 < H0 < H~2. For 0 < H~ < ~ a stripe pattern is stable. Detailed predictions agree very well with experiment.
RECENTLY much attention has been drawn to the formation and manipulation of cylindrical domains in the rare earth orthoferrites.’ These “bubble” domains are also foundininthin other 25 generally occuring platelets materials, with a perpendicular easy axis and in the presence of a (relatively high) external magnetic field also normal to the plate. The stability of isolated (non-interfacing) bubble domains was
characteristic lengths structure in the walls, domain configuartion: sotropy stant, constant, and
first3 investigated Kooy and recentlytheoretically extended in by detail by and Thiele.6 Enzconsideration was given to the interaction No
can be written as: ‘7
governing respectively the and the macroscopic K,.. is the uniaxial aniA0 is the exchange con-
The total magnetostatic energy in this model of a single reverse domain of volume v and wall area A in the presence of an external field H
0 u
compare our results with existing experimental data, The phenomenological model we employ to3’68 treatconsider the domains has appliedh elsewhere. We a plate of been thickness and infinite lateral dimensions and assign a surface energy a~ to the domain walls which are assumed to be everywhere perpendicular to the surface. Implicit in the model is the assumption that 2irM.~ =
K
‘-~sat
A
1 2 = — v[1 — N — H~]+ —~- (2) 87TM8 Here ~ is the energy of the plate with no reverse domains, N is the ~average)demagnetization factor of the domain, and fi 0 H0/4~M~. For cylindrical domains the factor N depends only on the ratio of the diameter d to the height h. Thus u1 is to. be minimized with respect to d/h for to determine of which H0 for the given ~/h which u the < 0;values i.e., for domain is stable.”3’6 When u 1 < 0, it is clear that a multiplicity of domains is favored with the equilibrium domain density limited by the interdomain interaction.
between bubbles which the might be and expected to influence substantially size distribution of the domains. In the following we discuss some of our theoretical studies of the interaction and
~
—
Of the possible “monatomic” space filling =
<< 1
(1)
lattices of cylinders, it can be shown that the lowest energy configuration is the triangular (closest that casepacking)~lattice. can be written as:The energy density in
Here M3 is the spontaneous 1 /2, and ~ = magnetization, are the = 2(A~/K,~)
1303
1304
MAGNETIC BUBBLE DOMAIN INTERACTIONS ULatticeU~at 2V 8nM~ —
(_~__)(i ‘~‘ceU
H0
—
____
m,n
2E~-)+(.~x_)2
____
q
(q~~y2)2
an increasing field as H 0 exceeds H~1,or, in a decreasing field as H 0 becomes less than He,. Rather, various artificial 1 ,5 means have been
=
J~(q~y/2) (1- e~x) x
Vol.8, No. 16
used to produce an abundance of bubbles, (e.g., the “cutting” of stripes with a magnetized needle). (3)
The fact that the bubble lattice does not
mn
where x = h/a, y = d/a, a is the 2/2 nearest is the unit neighbor distance, V~ = \/(3) a cell volume, and q~ = 477/~/(3) (m2 + mn + 02)’ 2• The energy tunction, equation (3) is to be minimized with respect to x and y to determine the stationary values of d and a for given H0 and ~ /h. Some of our results are shown on Fig. 1 and other results and conclusions are summarized qualitatively in the following. At H0 = 0, the cylindrical domain lattice is energetically less favorable 3(byThis roughly 1—5%) situation than the familiar stripe pattern. obtains for all H 0 ~ H~,(region I). At H~ = H~ (at roughly 1/3 the saturation field) the bubble lattice energy density equals the stripe energy density, and the bubble lattice is then the favored structure throughout region II, H~,~ H0 < H~2,where H~2 is defined by
~ H
~‘Lottice 11c2
—
Usat
=
u,
=
0. In region III,
0 < H~3,isolated bubbles may exist metastably; i.e., an energy barrier prevents their collapse. At H~ H~2 and h/e 4 (where the domain diameters are smallest; see Fig. 1) this2M~h3. barrierFor height is about in units of example, for 0.1 TmFeO 2ir 3 at h/~ = 4, the barrier is about 6 x 10~ergs/bubble. It should be noted that the same barrier inhibits the nucleation of bubble domains. Thus, one may expect that in the applied filed range from H~3 to somewhat less than H02 ,~ a sparse density of weakly interacting bubbles can be maintained while individual bubbles may be artificially created and destroyed, fixed in position, or moved about by external means with the expectation that none will appear or disappear
appear spontaneously when it is the lower energy structure and the fact that it has been observed at H 0 = 0 where it is not the lowest energy structure are complications related to the mechanism of the nucleation, the existence of wallcoercivity in non-ideal materials, and the fact that conversion of stripe arrays to bubble arrays involves the formation of (temporary), excess walls and hence an intermediate energy barrier. Discussion of these matters is beyond the scope of this letter. Our main purpose is to point out that the present calculations agree well with published data lattices in those have casesbeen where essentially regular bubble observed and the numerical values reported. Given the plate thickness, external magnetic field, and the values of 47r&15 and ~, the present model is used to predict the nearest neighbor distance a and domain diameter d which may be compared with observation. We have been able to find only three cases for which sufficient data were presented. Interestingly, the data are for three basically different magnetic materials. The results are shown in Table 1. The good agreement may be suprising in view of the methods resorted to, to produce the bubbles in these experiments. In this regard, our calculations show that over a wide range in H0, the bubble lattice “fits” the iso-field stripe array, i.e., the distance between stripes equals the distance between bubble rows. This situation is undoubtedly conductive to the production of a nearly equilibrium bubble array. A detailed presentation of our calculations 1
10
spontaneously.’
will be published eisewhere.
In practice, it is typically observed that the bubble lattice does not appear spontaneously in
Acknowledgements — we are indebted to L. Vredevoe for valuable discussion.
Vol.8, No.16
MAGNETIC BUBBLE DOMAIN INTERACTIONS I
~1~
I
I I
40
~
1305 I -
~m,
________
h/~
FIG. 1. The reduced critical fields as a function of the reduced plate thickness, and the corresponding critical bubble diameters. The fields are defined by: H 01, the bubble lattice energy density equals the stripe domain energy density; H02 , the bubble energy equals zero; and H03 , the energy barrier stabilizing isolated bubbles vanishes. The curve d0 is the metastable bubble diameter at H0 = 0. Curve d03 has been attributed to Thiele (see reference 1).
ic1ble 1. Comparison of observed and calculated values of bubble diameter and separation. Material BaFe12019 TmFeO 3 GdIG
*
Ref. 3 m 1 5,7
d(obs.)
*
4cm
(0.5— x lO 3x 101.0) ~ l. 4.5 x 103cm
a(obs.)* (7.0) x lO4cm
d(calc.) 1 x 104cm
a(calc.) 6.7 x lO4cmt
4.3 x 10~cm
1.26 x 102cm~
4.3 x 10~cm~
6.0 x 103cm
4.5 x 103cm §
6.1 x 103cm~
Average values in some cases. Includes correction for finite geometry of plate but none for (27TM. 2/K~) 0 (see reference 3). 9 Applied field not given in reference 1. Calculated results are best fit at U = 0.245.
§
Present calculated values based on
2.5 x i0~, roughly twice the value given in reference 7
=
which is thought to be in error due to computational errors in that work; see reference 7. REFERENCES 1.
BOBECK A.H., Bell Sys. Tech. J. 46, 1901 (1967); BOBEK A.H., FISCHER R.F., PERNESKI A.J., REMEIKA J.P. and Van UITERT L.G., IEEE Transactions on Magnetics, Mag. 5, 544 (1969).
2.
SHERWOOD R.C., REMEIKA J.P. and WILLIAMS H.J., J. appi. Phys. 30, 217 (1959).
3.
KOOY C. and ENZ U., Philips Res. Rep. 15, 7 (1960).
4.
MEE C.D., Contemp. Phys. 8, 385 (1967).
1306
MAGNETIC BUBBLE DOMAIN INTERACTIONS
Vol.8, No.16
5.
NEMCHIK J.M., J. appi. Phys. 40, 1086 (1969).
6.
THIELE., Bell Sys. Tech. J. 48, 3287 (1969).
7.
CHARAP S.H. and NEMCHIK J.M., IEEE Transactions on Magnetics, Mag. 5, 566 (1969). These authors have also derived equation (3) for the energy. However, we find that their numerical solution of the minimization of equation (3) is in error. A detailed discussion of the computational problem and the source of errors will be discussed in a full article (reference 10).
8.
For a review of the theory of ferromagnetic domains, see KITTEL C., Rev, mod. Phys. 21, 541 (1949).
9.
The exact lower field limit of stability of an isolated bubble has been determined by THIELE, reference 6.
10.
CAPE J.A. and LEHMAN G.W., manuscript in preparation.
Nous avons consideré l’interaction entre des domaines magnitiques de geometrie cylindrique (“Bubbles”) et prédit une configuration de reseau triangulaire stable dans une champ appliqué, H 01 < H0 ‘102. Pour 0 < H0 < H01 un reseau de bandes est stable. Les predictions detaillés son en bon accord avec les resultats d’experience.