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Conical bubbles
Journal of Magnetism and Magnetic Materials 124 (1993) 355-367 North-Holland
Conical bubbles A. Thiaville and K. PBtek ’ CNRS-Universitk
Paris Sud, Laboratoire
de Physique des Solides, Bit. 510, 91405 Orsay, France
Received 30 November 1992
In magnetic garnet films with perpendicular anisotropy, cylindrical domains called bubbles are well known characteristic domain structures. Using the anisotropic dark field technique in an optical microscope, we have investigated the bubble wall contrast as well as that of vertical Bloch lines contained in them. It is found that bubbles have in fact a slightly conical shape, with a tilt of the order of degrees, depending in sign and magnitude on the sample. We interpret it as a consequence of a variation in the sample parameters along its thickness. A numerical evaluation of tilt is proposed in the case of a gradient of wall energy density. Also, a simplified optical calculation of light diffraction at a wall is presented and is used for fitting experimental data.
1. Introduction
Magnetic bubbles are cylindrical domains with circular cross sections, appearing in platelets or thin films of magnetic samples with perpendicular uniaxial anisotropy [ll. The domains are cylinders (with walls parallel to the easy axis) in order to avoid magnetic poles on the wall. They can be easily seen in an optical polarizing microscope, due to Faraday rotation of light polarization. A few years ago, a variation of this technique, the anisotropic dark-field technique, was shown to also provide information on the wall structure [21. Although the first pictures of bubbles in anisotropic dark field already contained the features which we shall discuss here, it is only because of the recent understanding of the information provided by anisotropic dark-field imaging [3] that we cquld interpret them. As will be made
quantitatively clear below, anisotropic dark field is an observation technique that is extremely sensitive to the tilt of a wall. Here, a coherent explanation of the bubble picture as seen in an anisotropic dark field experiment is presented, based on a slightly conical bubble shape [4]. Also, the peculiar contrast of the lines sitting on a bubble wall (compared to lines on walls of stripe domains) can be evaluated according to the same hypothesis. The organization of the paper is as follows. After a description of the experiment, the results on bubble walls are discussed in section 3. Then, based on their interpretation, the results pertaining to lines in the bubble wall are analyzed in section 4. Calculations of the diffracted amplitude by a wall and of the tilt of the bubble wall, are presented in appendices A and B, respectively.
2. Experiment Correspondence
to: Dr. A. Thiaville, CNRS-Universitt
Paris de Physique des Solides, Bit. 510, 91405
Sud, Laboratoire Orsay, France. ’ On leave from the Institute of Physics, Czech Academy of Sciences, Na Slovance 2, 18040 Prague 8, Czech Republic. 0304-8853/93/$06.00
Figure 1 describes schematically the experimental arrangement. A high numerical aperture (0.9) condenser, fitted with an easily movable
0 1993 - Elsevier Science Publishers B.V. All rights reserved
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3. Images of bubble walls not containing
condenser diaphragm
Fig. 1. Principle
of the anisotropic
dark field method.
diaphragm in its back focal plane, is used to produce an inclined light beam. We will describe its inclination by the numerical aperture NA = 12sin 0 (n is the refractive index, 19is the angle relative to the axis), a quantity invariant by refraction on horizontal surfaces. In order to realize dark-field conditions, the incident numerical aperture has to be chosen larger than the objective’s acceptance capacity (here 0.4). Different bubble garnet films were investigated. As the results were qualitatively the same, we will show here results pertaining to a single sample that was utilized in previous studies [5]. The parameters of importance here are the sample thickness h = 7.5 km and stripe domain width w = 7.1 Km. Bubble lattices were prepared by in-plane field saturation, in a separate electromagnet equipped with a microscope. The bubbles contained two winding lines, as expected and checked by direct observation. By applying small in-plane fields of appropriate orientation, the position of the lines could be controlled. Their position did not change after the removal of the in-plane field (this point is discussed at length in ref. [6]).
lines
Figure 2(a) shows a bright field image of the bubble lattice, with slightly uncrossed polarizers in order to reveal both types of domains by the Faraday effect. The corresponding anisotropic dark field image is displayed in fig. 2(b) with the polarizer and analyzer removed. The bubble picture consists of two crescents only, because the walls which are nearly parallel to the incident beam (the beam is here inclined along the horizontal axis of the drawing) do not diffract. The picture has been contrast-inverted because it is easier to show black spots on a bright background than the converse. In fig. 2(b) the experimental fact at the origin of this work is clearly seen, namely, the two crescents have different intensities. A more detailed investigation of this difference yields the following conclusions: (i) Which crescent, left or right, is more intense does not depend on the bubble magnetization, as shown by the figures, which were chosen to contain both bubble polarities. (ii> However, if the sample is turned upside down, the more and less intense crescents change sides. This may be seen by comparing figs. 2(d,e,f) with figs. 2(a,b,c), respectively. (iii> If the numerical aperture (NA) of the incident beam is increased, the relative intensity of the crescents changes. For example, one reversal occurs between NA = 0.6 and 0.77 (compare figs. 2(e) to (f), and (b) to Cc), respectively). Another reversal occurs for NA in excess of 0.77, but the diffracted intensity then becomes too small for proper picture reproduction. (iv> Under the same conditions (up or down, first NA region of the dark field) which crescent appears the brighter may change according to the samples. Samples where the two crescents exhibit the same intensity also exist. From these results it is clear that the up and down sections of the bubbles are different. One obvious reason is the presence of the substrate on one surface only, but this is not likely to affect the picture optically. Rather, we have to accept that the bubbles really have different up and down parts. It is known that upon wall tilting the diffracted intensity changes and, more precisely,
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that in the first region of the dark field, the diffracted intensity increases when the wall tilts towards the incident beam (131;see also appendix
357
A). The idea of conical bubbles comes naturally. Inspection of the insets in fig.. 2 shows that it accounts for the experimental facts qualitatively.
Fig. 2. Images of a bubble lattice with the two bubble polarities. (a,b,c): Magnetic epilayer downside; (d,e,f): epilayer upside. (a and d): Bright field pictures with the same polarizer setting. The insets exhibit the corresponding structures. (b and e): Dark field picture at NA = 0.6. (c and f): Idem, at NA = 0.77. The dark field pictures are contrast-inverted for better reproduction.
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In appendix A, a calculation of the diffracted intensity by a planar tilted wall is made. As the bubbles are of large diameter, it may be applied to the two crescents of the bubble, supposing only that they are tilted by angles fa. Using the same values which fit the straight wall intensity data of ref. [3], namely refraction index IZ= 2.2, wavelength of light A = 0.59 pm, average numerical aperture of the rays contributing to the picture NA, = 0.35, and supposing (Y= -+ 2”, we get the two curves denoted R (right crescent) and L (left crescent) in fig. 3. In this picture, as in the following, the case of the film oriented downside will be considered. Also, the incident beam is travelling upwards and is inclined toward the right side. Two contrast equalities occur at values of NA called E, and E,. In that case E, = 0.6823 and E, = 0.7755. These theoretical predictions are in qualitative agreement with the experimental data. Trying to be more quantitative, we would like to extract from the experimental data a value of (Y.The first idea that comes to mind is to measure E, and E,, but here we face a difficulty that is not apparent in the picture of fig. 2. The values of E, and E, vary considerably from bubble to bubble. In fig. 4, the theoretical dependence of E, on (Y
300
[
0
, *I 0.6’5 u5 0.7
I
0.55 I
0.6 /
.L
I
I
+
I
0.75 E2 0.8 ‘N.A. J
I
i
I
Dark field image COllIrnSl lnvened Fig. 3. Calculation, versus NA, of diffracted intensities by the left (L) and right (R) walls of a bubble in the conditions of fig. 2 with the film downside. Two crossing points denoted E, and E, occur. Tilt angle used (Y= f 2”.
2 1 0
& %.oo 0.65O 0.66 0.67 0.68 0.69
0 7 0.71 ‘N.A.
Fig. 4. Calculated dependence of El on (Y within the model used for fig. 3. The experimental points are shown as open symbols. See text for comments.
is shown together with experimental points. Comparing both, one could have the impression that the wall tilt is nearly random. The picture in fig. 2, however, tells the contrary, simply because all bubbles show the same type of contrast away from the E, and E, zones, Therefore, the main noise in the experimental data does not originate from (Y. In ref. [7], a variation of wall contrast along a straight wall without line had already been seen, and tentatively attributed to small deformations of the wall. The present experiment shows that random wall tilt is not at stake. The hypothesis of sample thickness ofluctuations can also be rejected as a mere 50 A roughness was measured by means of a Dektak profilometer, which would give rise to a 10m4 fluctuation in the E, value. A simple effect can nevertheless be seen to exacerbate the influence of the noise source in this measurement. E, and E, occur near the aperture where the wall diffracted intensity is at a minimum. Therefore, any noise source which is additive to the wall diffracted intensity will disturb deeply the measurement of NA at which two crescents show equal intensities. If we restrict ourselves to the mean values of the experimental results, a tilt angle (Y= 1.7” is found for the bubbles of this sample. It now remains to explain why bubbles assume this conical shape, independently of their magnetization (up or down). A side question is also why this effect is not seen on stripe domains. As all
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experiments were performed on liquid phase epitaxial films, we examine the hypothesis of a variation in material parameters (exchange constant A, anisotropy constant K,, magnetization 41rM,) along the sample thickness. Characterization techniques have shown that they indeed vary 181. Consider first a bubble lattice. From the material parameters, the equilibrium bubble radius and bubble separation can be computed 111. Thus, schematically, different bubble radii and separations are desired at the two surfaces. The matching condition between the structures at the two surfaces then requires an identical lattice, but the bubble radius remains free and can vary between the two surfaces, resulting in conical bubbles. The fact that the bubble polarity has no influence on its shape is a consequence of the varying of scalars (the material parameters) only. Appendix B describes a calculation for an isolated bubble in the case where the sole wall energy is changing along the thickness. We find that a 170% variation in wall energy is needed to create a 1” tilt for the sample considered. This value may seem extremely large, but is, after all, not so unreasonable. We can compare these results with those obtained by ferromagnetic resonance (the technique of ref. [8]). Here, as the damping parameter is large and the sample is thick, individual modes cannot be resolved, but a (inhomogeneous, proportional to frequency) line broadening occurs. In order to separate the homogeneous (damping, independent of frequency) and inhomogeneous (from magnetic constant gradients) contributions, measurements were made at different frequencies. The results are shown in fig. 5, and display a sizeable inhomogeneous broadening. The value in the parallel geometry (fields applied in the sample plane) is roughly one-half of the perpendicular geometry figure, because of the different ways the anisotropy field enters the resonance equation (see, for example, ref. [SD. Taking simply the value of the inhomogeneous broadening in a perpendicular field to be the width of the anisotropy field distribution, and comparing it with the mean anisotropy field, a 40% variation in anisotropy is found (which would turn into a 20% variation in wall energy). This estimate is conservative because the non-spin-
8 ._
500
359
t
0
1
2
4
6 f (GHz)
8
10
12
Fig. 5. Peak-to-trough linewidth of the derivative of FMR absorption, measured at various frequencies under perpendicular and parallel fields.
wave modes do not cover all the width of the anisotropy field dispersion [81. A more detailed comparison is hindered by the large value of the damping parameter, but at least a wide span for wall energy variations can be accepted. Consider now the case of stripe domains. First, we recall that in zero bias field, where the widths of up and down domains are the same, no intensity difference between the two walls can be detected (fig. 6b). The reason is simple. A parallel stripe lattice has only one parameter in zero field, the stripe width, which is required to remain steady by the matching conditions between the two surfaces. On the contrary, as shown in figs. 6(a) and (c), the wall diffracted intensities change when a bias field shrinks one type of domain and expands the other (indeed, a comparison of figs. 6(a) and (cl shows that wall tilting is such that the shrunken domain takes the same shape: as for bubbles, domain polarity has no influence). This is because the amount of shrinkage becomes a free parameter of the lattice. Note that in this case, wall energy variations will play no role. Thus fig. 6 demonstrates that a variation in 47rM, occurs. Comparing with the corresponding fig. 2(e), we see that bubble and stripe tilts are of the same sign, putting the smaller domain on a cone with apex upward. That the bubble tilt appears larger means that the wall energy variation seems to exist and adds to the magnetization variation. But being more quantitative is experimentally difficult.
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lines along the wall should appear. Such an effect has not yet been observed, however.
4. Images of lines in bubble walls
Fig. 6. Images of parallel stripes in dark field (NA = 0.61, with epilayer up. (a) Bias field - 50 Oe; (b) zero applied bias field; (c) bias field +50 Oe (the sample magnetization is 4~rM, = 185 G).
Finally, we outline a solution to the case of stripe domains in zero field. Realization of different stripe domain widths at the two surfaces can be reached by losing invariance along the wall direction, as shown in fig. 7. This structure was mentioned by Goodenough [9] a long time ago in the case of magnetoplumbite. In anisotropic dark field, an illusion of the presence of a chain of
Using a small in-plane field, lines can be moved to the visible parts of the wall (the crescent) or to the hidden parts. Thus substraction of two images, one with, the other without lines could be used to enhance their contrast. As already shown in ref. [5], the line contrast (defined as the excess or reduction in the intensity diffracted by the wall) changes with numerical aperture of the incident beam. This is reproduced quite well by the calculation of appendix A, as shown in fig. A2. The fact that the bubble wall is tilted (and that the two facing crescents have opposite tilts) leads to the difference, versus numerical aperture, between the contrast behaviour of lines belonging to opposite crescents. We now explore this effect in more detail. Figure 8 shows line images for three inclinations of the incident beam (travelling from down to up and left to right). Figures 8(a,b,c) and 8(d,e,f) were taken with the magnetic film downside and upside, respectively. The same inplane field was used to bring the lines to the desired position. The situation and labels used are summarized by drawings. Let us concentrate on the picture taken with the film downside. In the first region of anisotropic dark field contrast (NA = 0.6, fig. 8a>, two dark and two bright lines can be seen (this difference in line contrast oc-
Fig. 7. Proposed undulation deformation of stripe domains in zero applied field, in the case of a lower wall energy (of higher magnetization) at the top surface.
A. TXauille, K. Patek / Conical bubbles
curs when the bubble polarity changes, because the wall magnetizations are the same). Going to higher incidence angle (NA = 0.67, fig. 8b), one
Ld
Lb
361
line denoted Ld changes contrast first. Then, at a still higher numerical aperture (NA = 0.77, fig. 84, Ld again changes contrast.
Lb Rb
d
Fig. 8. Dark field images of lines in bubbles, on the same bubble lattice as in fig. 2. (a,b,c): Epilayer downside; (d,e,f): epilayer upside. Wall structures and labels are explained by drawings. Note the contrast reversals that occur at different NAs for the different lines, an effect of bubble wall tilt.
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362 300 ” ? 250 % .g
200
.g
1.50
B z
100
$ %
50 0
0.55 Fig. 9. (a) Computation
1
0.58
0.65
0.1 N.A.
of walls and dark lines contrast corresponding to fig. 8, epilayer downside (bubble tilt angle - 3” assumed). (b) Same as (a) for the bright lines (line tilt angle + 3”).
Moreover, Rd and Lb have also experienced their first contrast reversal at that time, whereas Rb still has not. Figure 9 displays the predictions of our model. All the parameters are fixed by the previous experiment. Figure 9(a) refers to the dark lines discussed above. Their calculated contrast reversals are consistent with experimental results. Figure 9(b) pertains to bright lines (still with the film downside). Again the expected contrast reversal occurs at a value consistent with experiment. We thought originally that experimenting with lines would allow for a more precise determination of the bubble wall tilt angle LY,but in fact,
tilt ’
0.6
I
0.6
I
0.62
I
0.64
I
0.66 N.A.
Fig. 10. Calculated dependence of the first line contrast reversal (that of line Ld in fig. 8a) on the wall tilt angle (Y, and span of experimental data.
0.75
0.8
wall tilt angle
0.85 f 2”, line
the same main problem arises. For example, fig. 10 shows the expected dependence on wall tilt CY, deduced from the position of the first observed line contrast reversal (line Ld crossing the wall L curve>. The experimental points are again too scattered for a precise determination of CX. We nevertheless wish to stress that all observed phenomena can be explained under the present conical bubble hypothesis. The influence of noise, however, would have to be eliminated to fully test this hypothesis. One result of this study is that the wall tilt is a well defined quantity, definitely less noisy than the measurements of contrast reversals would suggest at first.
5. Conclusion
In conclusion, we propose the idea of a small departure from the ideal bubble structure, giving rise to what we call conical bubbles. This weak tilt angle (of some degrees) could be only revealed through the highly sensitive anisotropic dark field observation technique. Thus, dark field observation of bubble (and stripe) domains is a rapid (but indirect) evaluation of the uniformity of the sample along its thickness, which can be useful as a tool with which to characterize the perfection of the sample. In the case of a strong
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nonuniformity some peculiar domain shapes may occur, as drawn for example in fig. 7, and this should be borne in mind when observing domains and trying to understand their shapes. At the present stage of investigation, we see no strong effect on domains and domain wall properties directly related to the tilt.
main. The wall structure and thickness is neglected here (the thickness effect was investigated in ref. [5]). In the computation of the diffracted amplitude, multiple scattering is neglected (the so-called Born approximation). Denoting k = 2r/A as the light wavevector in a vacuum, the diffracted amplitude reads:
Appendix A: Light diffraction
A =lrn sgn(x-x(z)) --m
by a domain wall
The problem of calculating the anisotropic dark field image of a (tilted) wall is complex when the sample thickness is non-negligible when compared to the wavelength: the optical transfer function in three dimensions is much more complicated than its two-dimensional equivalent (Airy function) [ll]. We therefore propose an approximate treatment to evaluate only the wall diffracted intensity. The calculation thus reduces to a plane wave problem: an incoming wave (angle of incidence ei, see fig. Al) gives rise to diffracted waves (inclination 0,). The amplitude of a diffracted wave representative of those entering the objective lens and contributing most to the image is evaluated. This amplitude is computed under the Fraunhofer approximation, because the objective is very far (in units of wavelength) from the sample. Further approximation treats the magneto-optical effect as a pertubation. In this picture the sample is taken as a medium of uniform refractive index n, filled with scattering centers. For a linearly polarized incident plane wave, these centers radiate perpendicularly polarized light with a positive sign for an up domain and a negative sign for a down do-
X
h’2
/ -(h/2)
+z(cos
dx
exp{ikn[x(sin Bi - cos
ei - sin 0,)
e,)]} dz.
(Al)
x integration runs to infinity, but would be in practice smoothly cut off, by letting all the possible 0, waves that can enter the objective interfere in the image plane, at the resolution spot of the objective. To remove the sign function and reduce the problem to an integration along the wall, we integrate by parts over x and ignore the terms at infinity. A = 2/_h:h;2,exp{ikn[ _r( z)(sin +Z(COS
/(ikn(sin
ei -
cos
ei - sin e,)
e,)]}
ei - sin e,)} dz.
(A2)
This procedure is akin to the boundary diffraction wave technique of diffraction theory [ill. Specializing (A2) to the case of a straight wall making an angle (Y with the film normal, we get A=
2h
ikn(sin 8i - sin e,)
.-
sin X X
’
(A31
with X= kni
[cos Bi - cos e, + tgcu(sin Bi - sin e,)]. (Ad)
X
t
l
Fig. Al. Parameters
d
-h/2
and coordinates used in wall diffraction calculation.
The behaviour of (A3) is well known, displaying a first zero value at X = rr. The case LY= 0 was discussed in ref. [51, where it was shown that the wall diffracted intensity versus 19~could be reproduced by such a formula.
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100 SO 0 0.55
I
I
0.6
0.65
I
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I
I
0.75
0.8
0.85
0.55
0.6
0.65
0.7
0.75
0.8
0.85
N.A. N.A. Fig. A2. Fit of wall (a) and lines (b) contrast variation versus NA, using relation (A31 for the diffracted amplitude. Baseline value ( = 50) and scaling factor adjusted on the wall curve. Tilt angle f 3” used for the lines. Points: experimental points, dashed curve: fifth-order polynomial fit for experimental points, full curve: calculated variation. Optical parameters are given in the text.
The prefactor in (A3) giving an intensity fall off with increasing ei, results in fact from many refractions at lenses and the sample interfaces, and is therefore complicated. A prefactor derived from transmittivity measurement of the lens system was used in ref. [5]. Here we simply keep the prefactor of GUI, since it leads to reasonable results. Moreover, as we refer to comparison of intensities in different cases, at the same angle Bi, the influence of the prefactor is minor for our purposes. Figure A2 shows the fit of experimental data [5] with formula (A3). Parameters are A = 0.59 urn, n = 2.2, h = 7.5 pm, n sin Be = 0.35. The background intensity and a scaling factor were adjusted to reproduce the wall values. To reproduce line contrasts, a local tilt LY= +3” was chosen. Despite the simplicity of the model, the agreement is quite satisfactory. Therefore, this model may serve as a guide to understanding. It explains also why the anisotropic dark field technique is so sensitive to wall tilt.
Appendix B: Calculation wall energy gradient
of bubble wall tilt due to
Consider an isolated bubble (it may require a bias field to be stable) in a sample where the wall surface energy, but not the magnetization, varies
with the thickness. For a Bloch wall, this means a variation of the exchange constant and/or the uniaxial anisotropy constant. Starting from the equilibrium configuration in the uniform material (radius r>, a small gradient in wall surface energy and a small tilt angle (Yare introduced. A variation of the bubble total energy occurs, which should be minimal at the resulting CX.Denoting by u the wall surface energy, and VU its gradient, the energy variation of the wall energy term reads to 0(a2):
AE,=
,rrh3
-
6
Vua + rrrhoa2.
(Bl)
The first term in (Bl) is the coupling of wall tilt to wall surface tension gradient, which will be negative. The second, which is positive, acts as a tilt limit because of the wall surface increase. If an applied field is present, a contribution is expected from the field energy due to a change in the bubble volume, but it is usually negligible (see the remark after (BlO) combined with the shrinking field present in a bubble lattice of a usual material, H N 0.5 M, (1)). The last term required, linked to the demagnetizing field energy, is the most difficult to evaluate. We shall use the magnetic charge formalism.
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the distance between two such lines, a pair energy reading:
t
/h
BUBBLE:CUTVIEW
eb( x) = 4MS2a2 + MS
ORIGINAL (VALUES
_:I:III--I::I:::I:
-MS
-2Ms
-2Ms
h2
2x+
2G-Z
+Ms
-MS
+Ms
h2
CHARGE DISTRIBUTION PER UNIT SURFACE)
-2@-Si2x.
w I
+Ms
-2Ms
-2Ms
Fig. Bl. Calculation of conical bubble demagnetizing field energy variation. (a) Cross sections of the magnetization directions; (b) the original (cylindrical bubble) magnetic charges distribution; (c) the additional charge arising from wall tilt.
This energy becomes infinite (as x-l) when x goes to zero, because of the lumping into a point charge at the surfaces. The divergence which would arise when integrating along the bubble wall is treated as follows. The integration with (B3) will stop at x =p(h/2)a (p times the real extension of the charge at the surface). The integration in the zone close to zero uses a form with spread surface charges, where the h2/2x term of (B3) becomes
The charge density is -div M and the demagnetizing energy reads:
d3r d3rr.
(B2)
As shown in fig. Bl, there are original charges and additional charges appearing because of the tilt. If the additional charges appearing on the sample surfaces are lumped to a 6 function, then all additional charges appear to be proportional to 1~. AE, would then comprise an interaction term between original and additional charges proportional to (Y, and a self-energy term of the additional charges, proportional to CY~.The interaction term must be zero, because by up-down symmetry positive and negative (Y give the same result. This also ensures that bubbles in a uniform material are not conical. The self-energy term can be computed as arising from a collection of linear segments distributed along the wall. In a first approximation they are assumed to be vertical, bearing a linear charge density +2M,(w along their length and two -M,crh point charges at their ends. Integration of (B2) yields, if x is
-2&jZ.,,
VW
.
I
Finally, AE,
= i2Trk2reD(2r
sin: h2
= 16nM2r2cx2 zlns
)rd0
W)
8br (alh
+4r
h2
+
2W ,
P)
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where K and E are complete elliptical integrals of first and second kind [12], defined as
E(m)
= j”‘*deJ1_Msin’sdf3.
(B7)
0
0.5
is a number coming from the use of formula (B4) in the range x
b
0
Fig. B2. Plot of the bubble wall tilt angle a versus wall energy variation ratio, derived from eq. (B9).
w
+P(P- Vilfp2).
b is infinite for p = 0 (as seen before) and goes quickly to b = 3/2 + In 2 = 2.1931 when p grows. This limiting value will be used. Deriving with respect to (Y the total energy variation AE, obtained by summing (Bl) and (B6), one gets IaI(ln
Ial-b+c)
Trl h1Vul = --zp u
.
w9
The sign of (Yis such that the head of the cone is in the direction of larger wall energy density u. As usual, the characteristic length I is denoted by 1 = 0/47rM,2, and c is a number depending on 1, h, and r according to 1 +2-x--
+lnk
IT1
16r2 h2
over, this calculation was restricted to a conical shape, but that approximation is thought to be noncritical. From the experiments and from this calculation, it appears that the gradient of magnetization may be the main cause of the tilt. We do not propose here a calculation of this effect, because of the complication of the magnetostatic calculation: the surface charges on the wall are non-constant, and volume charges exist in the domains.
Acknowledgements We thank J. Ben Youssef from L.M.M.M. (Meudon, France) for performing the ferromagnetic resonance measurements at various frequencies, using the strip technique [lo]. J. Miltat contributed many interesting suggestions and comments, which are gratefully acknowledged.
(BlO) References
(c decreases by (h/24r) (H/M,) for an applied field shrinking the bubble). In the case of the standard bubble material, where h = 2r = 10 1, one finds c = 1.1289. Figure B2 plots the solution of (B9) with b and c given above. A tilt angle of 1” is obtained when the relative wall energy increases from 0.15 to 1.85 between the two surfaces. We therefore find that tilts of the order of 1” require quite large variations in wall energy. On the other hand, we did not investigate nonlinear variations of U. More-
[l] See, for example, Magnetic bubbles by A.H. Bobeck and E. Della Torre, in: Selected Topics in Solid State Physics, vol. XIV, ed. E.P. Wolfarth (North-Holland, Amsterdam, 1975). [2] A. Thiaville, L. Arnaud, F. Boileau, G. Sauron and J. Miltat, IEEE Trans. Magn. 24 (1988) 1722. [3] A. Thiaville, J. Ben Youssef, Y. Nakatani and J. Miltat, J.
Appl. Phys. 69 (1991) 6090. [4] In the early days of the bubble memories, published) observed so-called half-bubbles, analyzed by W.J. DeBonte, Bell Syst. Tech. 1933. Bobeck attributed their existence to
Bobeck (unwhich were J. 51 (1972) gradients of
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M, or (T. But DeBonte’s analysis chooses a model where a sole reduction in bubble height is taken into account disregarding any tilted wall effect. 151 A. Thiaville and J. Miltat, J. Appl. Phys. 68 (1990) 2883. [6] K. Patek, R. Gemperle, L. Murtinova and J. Kacier, J. Magn. Magn. Mater. 123 (1993) 223. [7] A. Thiaville and J. Miltat, IEEE Trans. Magn. 26 (1990) 1530. [8] B. Hoekstra, R.P. van Stapele and J.M. Robertson, J. Appl. Phys. 48 (1977) 382.
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[9] J.B. Goodenough, Phys. Rev. 102 (1956) 356. [lo] H. Le Gall, N. Vukadinovic, 0. Navarro and J. Ben Youssef, in SEE 89, JournCes d’Etudes Ceramiques en Hyperfrtquences et Nouvelles Applications, (SEE, Paris, 1989) p. 145; N. Vukadinovic, These UniversitC Paris VI, (1988). [Ill M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1964). [12] M. Abramowitz and I.A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
Conical bubbles A. Thiaville and K. PBtek ’ CNRS-Universitk
Paris Sud, Laboratoire
de Physique des Solides, Bit. 510, 91405 Orsay, France
Received 30 November 1992
In magnetic garnet films with perpendicular anisotropy, cylindrical domains called bubbles are well known characteristic domain structures. Using the anisotropic dark field technique in an optical microscope, we have investigated the bubble wall contrast as well as that of vertical Bloch lines contained in them. It is found that bubbles have in fact a slightly conical shape, with a tilt of the order of degrees, depending in sign and magnitude on the sample. We interpret it as a consequence of a variation in the sample parameters along its thickness. A numerical evaluation of tilt is proposed in the case of a gradient of wall energy density. Also, a simplified optical calculation of light diffraction at a wall is presented and is used for fitting experimental data.
1. Introduction
Magnetic bubbles are cylindrical domains with circular cross sections, appearing in platelets or thin films of magnetic samples with perpendicular uniaxial anisotropy [ll. The domains are cylinders (with walls parallel to the easy axis) in order to avoid magnetic poles on the wall. They can be easily seen in an optical polarizing microscope, due to Faraday rotation of light polarization. A few years ago, a variation of this technique, the anisotropic dark-field technique, was shown to also provide information on the wall structure [21. Although the first pictures of bubbles in anisotropic dark field already contained the features which we shall discuss here, it is only because of the recent understanding of the information provided by anisotropic dark-field imaging [3] that we cquld interpret them. As will be made
quantitatively clear below, anisotropic dark field is an observation technique that is extremely sensitive to the tilt of a wall. Here, a coherent explanation of the bubble picture as seen in an anisotropic dark field experiment is presented, based on a slightly conical bubble shape [4]. Also, the peculiar contrast of the lines sitting on a bubble wall (compared to lines on walls of stripe domains) can be evaluated according to the same hypothesis. The organization of the paper is as follows. After a description of the experiment, the results on bubble walls are discussed in section 3. Then, based on their interpretation, the results pertaining to lines in the bubble wall are analyzed in section 4. Calculations of the diffracted amplitude by a wall and of the tilt of the bubble wall, are presented in appendices A and B, respectively.
2. Experiment Correspondence
to: Dr. A. Thiaville, CNRS-Universitt
Paris de Physique des Solides, Bit. 510, 91405
Sud, Laboratoire Orsay, France. ’ On leave from the Institute of Physics, Czech Academy of Sciences, Na Slovance 2, 18040 Prague 8, Czech Republic. 0304-8853/93/$06.00
Figure 1 describes schematically the experimental arrangement. A high numerical aperture (0.9) condenser, fitted with an easily movable
0 1993 - Elsevier Science Publishers B.V. All rights reserved
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/ Conical bubbles
3. Images of bubble walls not containing
condenser diaphragm
Fig. 1. Principle
of the anisotropic
dark field method.
diaphragm in its back focal plane, is used to produce an inclined light beam. We will describe its inclination by the numerical aperture NA = 12sin 0 (n is the refractive index, 19is the angle relative to the axis), a quantity invariant by refraction on horizontal surfaces. In order to realize dark-field conditions, the incident numerical aperture has to be chosen larger than the objective’s acceptance capacity (here 0.4). Different bubble garnet films were investigated. As the results were qualitatively the same, we will show here results pertaining to a single sample that was utilized in previous studies [5]. The parameters of importance here are the sample thickness h = 7.5 km and stripe domain width w = 7.1 Km. Bubble lattices were prepared by in-plane field saturation, in a separate electromagnet equipped with a microscope. The bubbles contained two winding lines, as expected and checked by direct observation. By applying small in-plane fields of appropriate orientation, the position of the lines could be controlled. Their position did not change after the removal of the in-plane field (this point is discussed at length in ref. [6]).
lines
Figure 2(a) shows a bright field image of the bubble lattice, with slightly uncrossed polarizers in order to reveal both types of domains by the Faraday effect. The corresponding anisotropic dark field image is displayed in fig. 2(b) with the polarizer and analyzer removed. The bubble picture consists of two crescents only, because the walls which are nearly parallel to the incident beam (the beam is here inclined along the horizontal axis of the drawing) do not diffract. The picture has been contrast-inverted because it is easier to show black spots on a bright background than the converse. In fig. 2(b) the experimental fact at the origin of this work is clearly seen, namely, the two crescents have different intensities. A more detailed investigation of this difference yields the following conclusions: (i) Which crescent, left or right, is more intense does not depend on the bubble magnetization, as shown by the figures, which were chosen to contain both bubble polarities. (ii> However, if the sample is turned upside down, the more and less intense crescents change sides. This may be seen by comparing figs. 2(d,e,f) with figs. 2(a,b,c), respectively. (iii> If the numerical aperture (NA) of the incident beam is increased, the relative intensity of the crescents changes. For example, one reversal occurs between NA = 0.6 and 0.77 (compare figs. 2(e) to (f), and (b) to Cc), respectively). Another reversal occurs for NA in excess of 0.77, but the diffracted intensity then becomes too small for proper picture reproduction. (iv> Under the same conditions (up or down, first NA region of the dark field) which crescent appears the brighter may change according to the samples. Samples where the two crescents exhibit the same intensity also exist. From these results it is clear that the up and down sections of the bubbles are different. One obvious reason is the presence of the substrate on one surface only, but this is not likely to affect the picture optically. Rather, we have to accept that the bubbles really have different up and down parts. It is known that upon wall tilting the diffracted intensity changes and, more precisely,
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that in the first region of the dark field, the diffracted intensity increases when the wall tilts towards the incident beam (131;see also appendix
357
A). The idea of conical bubbles comes naturally. Inspection of the insets in fig.. 2 shows that it accounts for the experimental facts qualitatively.
Fig. 2. Images of a bubble lattice with the two bubble polarities. (a,b,c): Magnetic epilayer downside; (d,e,f): epilayer upside. (a and d): Bright field pictures with the same polarizer setting. The insets exhibit the corresponding structures. (b and e): Dark field picture at NA = 0.6. (c and f): Idem, at NA = 0.77. The dark field pictures are contrast-inverted for better reproduction.
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In appendix A, a calculation of the diffracted intensity by a planar tilted wall is made. As the bubbles are of large diameter, it may be applied to the two crescents of the bubble, supposing only that they are tilted by angles fa. Using the same values which fit the straight wall intensity data of ref. [3], namely refraction index IZ= 2.2, wavelength of light A = 0.59 pm, average numerical aperture of the rays contributing to the picture NA, = 0.35, and supposing (Y= -+ 2”, we get the two curves denoted R (right crescent) and L (left crescent) in fig. 3. In this picture, as in the following, the case of the film oriented downside will be considered. Also, the incident beam is travelling upwards and is inclined toward the right side. Two contrast equalities occur at values of NA called E, and E,. In that case E, = 0.6823 and E, = 0.7755. These theoretical predictions are in qualitative agreement with the experimental data. Trying to be more quantitative, we would like to extract from the experimental data a value of (Y.The first idea that comes to mind is to measure E, and E,, but here we face a difficulty that is not apparent in the picture of fig. 2. The values of E, and E, vary considerably from bubble to bubble. In fig. 4, the theoretical dependence of E, on (Y
300
[
0
, *I 0.6’5 u5 0.7
I
0.55 I
0.6 /
.L
I
I
+
I
0.75 E2 0.8 ‘N.A. J
I
i
I
Dark field image COllIrnSl lnvened Fig. 3. Calculation, versus NA, of diffracted intensities by the left (L) and right (R) walls of a bubble in the conditions of fig. 2 with the film downside. Two crossing points denoted E, and E, occur. Tilt angle used (Y= f 2”.
2 1 0
& %.oo 0.65O 0.66 0.67 0.68 0.69
0 7 0.71 ‘N.A.
Fig. 4. Calculated dependence of El on (Y within the model used for fig. 3. The experimental points are shown as open symbols. See text for comments.
is shown together with experimental points. Comparing both, one could have the impression that the wall tilt is nearly random. The picture in fig. 2, however, tells the contrary, simply because all bubbles show the same type of contrast away from the E, and E, zones, Therefore, the main noise in the experimental data does not originate from (Y. In ref. [7], a variation of wall contrast along a straight wall without line had already been seen, and tentatively attributed to small deformations of the wall. The present experiment shows that random wall tilt is not at stake. The hypothesis of sample thickness ofluctuations can also be rejected as a mere 50 A roughness was measured by means of a Dektak profilometer, which would give rise to a 10m4 fluctuation in the E, value. A simple effect can nevertheless be seen to exacerbate the influence of the noise source in this measurement. E, and E, occur near the aperture where the wall diffracted intensity is at a minimum. Therefore, any noise source which is additive to the wall diffracted intensity will disturb deeply the measurement of NA at which two crescents show equal intensities. If we restrict ourselves to the mean values of the experimental results, a tilt angle (Y= 1.7” is found for the bubbles of this sample. It now remains to explain why bubbles assume this conical shape, independently of their magnetization (up or down). A side question is also why this effect is not seen on stripe domains. As all
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experiments were performed on liquid phase epitaxial films, we examine the hypothesis of a variation in material parameters (exchange constant A, anisotropy constant K,, magnetization 41rM,) along the sample thickness. Characterization techniques have shown that they indeed vary 181. Consider first a bubble lattice. From the material parameters, the equilibrium bubble radius and bubble separation can be computed 111. Thus, schematically, different bubble radii and separations are desired at the two surfaces. The matching condition between the structures at the two surfaces then requires an identical lattice, but the bubble radius remains free and can vary between the two surfaces, resulting in conical bubbles. The fact that the bubble polarity has no influence on its shape is a consequence of the varying of scalars (the material parameters) only. Appendix B describes a calculation for an isolated bubble in the case where the sole wall energy is changing along the thickness. We find that a 170% variation in wall energy is needed to create a 1” tilt for the sample considered. This value may seem extremely large, but is, after all, not so unreasonable. We can compare these results with those obtained by ferromagnetic resonance (the technique of ref. [8]). Here, as the damping parameter is large and the sample is thick, individual modes cannot be resolved, but a (inhomogeneous, proportional to frequency) line broadening occurs. In order to separate the homogeneous (damping, independent of frequency) and inhomogeneous (from magnetic constant gradients) contributions, measurements were made at different frequencies. The results are shown in fig. 5, and display a sizeable inhomogeneous broadening. The value in the parallel geometry (fields applied in the sample plane) is roughly one-half of the perpendicular geometry figure, because of the different ways the anisotropy field enters the resonance equation (see, for example, ref. [SD. Taking simply the value of the inhomogeneous broadening in a perpendicular field to be the width of the anisotropy field distribution, and comparing it with the mean anisotropy field, a 40% variation in anisotropy is found (which would turn into a 20% variation in wall energy). This estimate is conservative because the non-spin-
8 ._
500
359
t
0
1
2
4
6 f (GHz)
8
10
12
Fig. 5. Peak-to-trough linewidth of the derivative of FMR absorption, measured at various frequencies under perpendicular and parallel fields.
wave modes do not cover all the width of the anisotropy field dispersion [81. A more detailed comparison is hindered by the large value of the damping parameter, but at least a wide span for wall energy variations can be accepted. Consider now the case of stripe domains. First, we recall that in zero bias field, where the widths of up and down domains are the same, no intensity difference between the two walls can be detected (fig. 6b). The reason is simple. A parallel stripe lattice has only one parameter in zero field, the stripe width, which is required to remain steady by the matching conditions between the two surfaces. On the contrary, as shown in figs. 6(a) and (c), the wall diffracted intensities change when a bias field shrinks one type of domain and expands the other (indeed, a comparison of figs. 6(a) and (cl shows that wall tilting is such that the shrunken domain takes the same shape: as for bubbles, domain polarity has no influence). This is because the amount of shrinkage becomes a free parameter of the lattice. Note that in this case, wall energy variations will play no role. Thus fig. 6 demonstrates that a variation in 47rM, occurs. Comparing with the corresponding fig. 2(e), we see that bubble and stripe tilts are of the same sign, putting the smaller domain on a cone with apex upward. That the bubble tilt appears larger means that the wall energy variation seems to exist and adds to the magnetization variation. But being more quantitative is experimentally difficult.
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lines along the wall should appear. Such an effect has not yet been observed, however.
4. Images of lines in bubble walls
Fig. 6. Images of parallel stripes in dark field (NA = 0.61, with epilayer up. (a) Bias field - 50 Oe; (b) zero applied bias field; (c) bias field +50 Oe (the sample magnetization is 4~rM, = 185 G).
Finally, we outline a solution to the case of stripe domains in zero field. Realization of different stripe domain widths at the two surfaces can be reached by losing invariance along the wall direction, as shown in fig. 7. This structure was mentioned by Goodenough [9] a long time ago in the case of magnetoplumbite. In anisotropic dark field, an illusion of the presence of a chain of
Using a small in-plane field, lines can be moved to the visible parts of the wall (the crescent) or to the hidden parts. Thus substraction of two images, one with, the other without lines could be used to enhance their contrast. As already shown in ref. [5], the line contrast (defined as the excess or reduction in the intensity diffracted by the wall) changes with numerical aperture of the incident beam. This is reproduced quite well by the calculation of appendix A, as shown in fig. A2. The fact that the bubble wall is tilted (and that the two facing crescents have opposite tilts) leads to the difference, versus numerical aperture, between the contrast behaviour of lines belonging to opposite crescents. We now explore this effect in more detail. Figure 8 shows line images for three inclinations of the incident beam (travelling from down to up and left to right). Figures 8(a,b,c) and 8(d,e,f) were taken with the magnetic film downside and upside, respectively. The same inplane field was used to bring the lines to the desired position. The situation and labels used are summarized by drawings. Let us concentrate on the picture taken with the film downside. In the first region of anisotropic dark field contrast (NA = 0.6, fig. 8a>, two dark and two bright lines can be seen (this difference in line contrast oc-
Fig. 7. Proposed undulation deformation of stripe domains in zero applied field, in the case of a lower wall energy (of higher magnetization) at the top surface.
A. TXauille, K. Patek / Conical bubbles
curs when the bubble polarity changes, because the wall magnetizations are the same). Going to higher incidence angle (NA = 0.67, fig. 8b), one
Ld
Lb
361
line denoted Ld changes contrast first. Then, at a still higher numerical aperture (NA = 0.77, fig. 84, Ld again changes contrast.
Lb Rb
d
Fig. 8. Dark field images of lines in bubbles, on the same bubble lattice as in fig. 2. (a,b,c): Epilayer downside; (d,e,f): epilayer upside. Wall structures and labels are explained by drawings. Note the contrast reversals that occur at different NAs for the different lines, an effect of bubble wall tilt.
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362 300 ” ? 250 % .g
200
.g
1.50
B z
100
$ %
50 0
0.55 Fig. 9. (a) Computation
1
0.58
0.65
0.1 N.A.
of walls and dark lines contrast corresponding to fig. 8, epilayer downside (bubble tilt angle - 3” assumed). (b) Same as (a) for the bright lines (line tilt angle + 3”).
Moreover, Rd and Lb have also experienced their first contrast reversal at that time, whereas Rb still has not. Figure 9 displays the predictions of our model. All the parameters are fixed by the previous experiment. Figure 9(a) refers to the dark lines discussed above. Their calculated contrast reversals are consistent with experimental results. Figure 9(b) pertains to bright lines (still with the film downside). Again the expected contrast reversal occurs at a value consistent with experiment. We thought originally that experimenting with lines would allow for a more precise determination of the bubble wall tilt angle LY,but in fact,
tilt ’
0.6
I
0.6
I
0.62
I
0.64
I
0.66 N.A.
Fig. 10. Calculated dependence of the first line contrast reversal (that of line Ld in fig. 8a) on the wall tilt angle (Y, and span of experimental data.
0.75
0.8
wall tilt angle
0.85 f 2”, line
the same main problem arises. For example, fig. 10 shows the expected dependence on wall tilt CY, deduced from the position of the first observed line contrast reversal (line Ld crossing the wall L curve>. The experimental points are again too scattered for a precise determination of CX. We nevertheless wish to stress that all observed phenomena can be explained under the present conical bubble hypothesis. The influence of noise, however, would have to be eliminated to fully test this hypothesis. One result of this study is that the wall tilt is a well defined quantity, definitely less noisy than the measurements of contrast reversals would suggest at first.
5. Conclusion
In conclusion, we propose the idea of a small departure from the ideal bubble structure, giving rise to what we call conical bubbles. This weak tilt angle (of some degrees) could be only revealed through the highly sensitive anisotropic dark field observation technique. Thus, dark field observation of bubble (and stripe) domains is a rapid (but indirect) evaluation of the uniformity of the sample along its thickness, which can be useful as a tool with which to characterize the perfection of the sample. In the case of a strong
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nonuniformity some peculiar domain shapes may occur, as drawn for example in fig. 7, and this should be borne in mind when observing domains and trying to understand their shapes. At the present stage of investigation, we see no strong effect on domains and domain wall properties directly related to the tilt.
main. The wall structure and thickness is neglected here (the thickness effect was investigated in ref. [5]). In the computation of the diffracted amplitude, multiple scattering is neglected (the so-called Born approximation). Denoting k = 2r/A as the light wavevector in a vacuum, the diffracted amplitude reads:
Appendix A: Light diffraction
A =lrn sgn(x-x(z)) --m
by a domain wall
The problem of calculating the anisotropic dark field image of a (tilted) wall is complex when the sample thickness is non-negligible when compared to the wavelength: the optical transfer function in three dimensions is much more complicated than its two-dimensional equivalent (Airy function) [ll]. We therefore propose an approximate treatment to evaluate only the wall diffracted intensity. The calculation thus reduces to a plane wave problem: an incoming wave (angle of incidence ei, see fig. Al) gives rise to diffracted waves (inclination 0,). The amplitude of a diffracted wave representative of those entering the objective lens and contributing most to the image is evaluated. This amplitude is computed under the Fraunhofer approximation, because the objective is very far (in units of wavelength) from the sample. Further approximation treats the magneto-optical effect as a pertubation. In this picture the sample is taken as a medium of uniform refractive index n, filled with scattering centers. For a linearly polarized incident plane wave, these centers radiate perpendicularly polarized light with a positive sign for an up domain and a negative sign for a down do-
X
h’2
/ -(h/2)
+z(cos
dx
exp{ikn[x(sin Bi - cos
ei - sin 0,)
e,)]} dz.
(Al)
x integration runs to infinity, but would be in practice smoothly cut off, by letting all the possible 0, waves that can enter the objective interfere in the image plane, at the resolution spot of the objective. To remove the sign function and reduce the problem to an integration along the wall, we integrate by parts over x and ignore the terms at infinity. A = 2/_h:h;2,exp{ikn[ _r( z)(sin +Z(COS
/(ikn(sin
ei -
cos
ei - sin e,)
e,)]}
ei - sin e,)} dz.
(A2)
This procedure is akin to the boundary diffraction wave technique of diffraction theory [ill. Specializing (A2) to the case of a straight wall making an angle (Y with the film normal, we get A=
2h
ikn(sin 8i - sin e,)
.-
sin X X
’
(A31
with X= kni
[cos Bi - cos e, + tgcu(sin Bi - sin e,)]. (Ad)
X
t
l
Fig. Al. Parameters
d
-h/2
and coordinates used in wall diffraction calculation.
The behaviour of (A3) is well known, displaying a first zero value at X = rr. The case LY= 0 was discussed in ref. [51, where it was shown that the wall diffracted intensity versus 19~could be reproduced by such a formula.
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A. Thiaville, K. Pdek
/ Conical bubbles 300 250 200 150
100 SO 0 0.55
I
I
0.6
0.65
I
0.7
I
I
0.75
0.8
0.85
0.55
0.6
0.65
0.7
0.75
0.8
0.85
N.A. N.A. Fig. A2. Fit of wall (a) and lines (b) contrast variation versus NA, using relation (A31 for the diffracted amplitude. Baseline value ( = 50) and scaling factor adjusted on the wall curve. Tilt angle f 3” used for the lines. Points: experimental points, dashed curve: fifth-order polynomial fit for experimental points, full curve: calculated variation. Optical parameters are given in the text.
The prefactor in (A3) giving an intensity fall off with increasing ei, results in fact from many refractions at lenses and the sample interfaces, and is therefore complicated. A prefactor derived from transmittivity measurement of the lens system was used in ref. [5]. Here we simply keep the prefactor of GUI, since it leads to reasonable results. Moreover, as we refer to comparison of intensities in different cases, at the same angle Bi, the influence of the prefactor is minor for our purposes. Figure A2 shows the fit of experimental data [5] with formula (A3). Parameters are A = 0.59 urn, n = 2.2, h = 7.5 pm, n sin Be = 0.35. The background intensity and a scaling factor were adjusted to reproduce the wall values. To reproduce line contrasts, a local tilt LY= +3” was chosen. Despite the simplicity of the model, the agreement is quite satisfactory. Therefore, this model may serve as a guide to understanding. It explains also why the anisotropic dark field technique is so sensitive to wall tilt.
Appendix B: Calculation wall energy gradient
of bubble wall tilt due to
Consider an isolated bubble (it may require a bias field to be stable) in a sample where the wall surface energy, but not the magnetization, varies
with the thickness. For a Bloch wall, this means a variation of the exchange constant and/or the uniaxial anisotropy constant. Starting from the equilibrium configuration in the uniform material (radius r>, a small gradient in wall surface energy and a small tilt angle (Yare introduced. A variation of the bubble total energy occurs, which should be minimal at the resulting CX.Denoting by u the wall surface energy, and VU its gradient, the energy variation of the wall energy term reads to 0(a2):
AE,=
,rrh3
-
6
Vua + rrrhoa2.
(Bl)
The first term in (Bl) is the coupling of wall tilt to wall surface tension gradient, which will be negative. The second, which is positive, acts as a tilt limit because of the wall surface increase. If an applied field is present, a contribution is expected from the field energy due to a change in the bubble volume, but it is usually negligible (see the remark after (BlO) combined with the shrinking field present in a bubble lattice of a usual material, H N 0.5 M, (1)). The last term required, linked to the demagnetizing field energy, is the most difficult to evaluate. We shall use the magnetic charge formalism.
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365
the distance between two such lines, a pair energy reading:
t
/h
BUBBLE:CUTVIEW
eb( x) = 4MS2a2 + MS
ORIGINAL (VALUES
_:I:III--I::I:::I:
-MS
-2Ms
-2Ms
h2
2x+
2G-Z
+Ms
-MS
+Ms
h2
CHARGE DISTRIBUTION PER UNIT SURFACE)
-2@-Si2x.
w I
+Ms
-2Ms
-2Ms
Fig. Bl. Calculation of conical bubble demagnetizing field energy variation. (a) Cross sections of the magnetization directions; (b) the original (cylindrical bubble) magnetic charges distribution; (c) the additional charge arising from wall tilt.
This energy becomes infinite (as x-l) when x goes to zero, because of the lumping into a point charge at the surfaces. The divergence which would arise when integrating along the bubble wall is treated as follows. The integration with (B3) will stop at x =p(h/2)a (p times the real extension of the charge at the surface). The integration in the zone close to zero uses a form with spread surface charges, where the h2/2x term of (B3) becomes
The charge density is -div M and the demagnetizing energy reads:
d3r d3rr.
(B2)
As shown in fig. Bl, there are original charges and additional charges appearing because of the tilt. If the additional charges appearing on the sample surfaces are lumped to a 6 function, then all additional charges appear to be proportional to 1~. AE, would then comprise an interaction term between original and additional charges proportional to (Y, and a self-energy term of the additional charges, proportional to CY~.The interaction term must be zero, because by up-down symmetry positive and negative (Y give the same result. This also ensures that bubbles in a uniform material are not conical. The self-energy term can be computed as arising from a collection of linear segments distributed along the wall. In a first approximation they are assumed to be vertical, bearing a linear charge density +2M,(w along their length and two -M,crh point charges at their ends. Integration of (B2) yields, if x is
-2&jZ.,,
VW
.
I
Finally, AE,
= i2Trk2reD(2r
sin: h2
= 16nM2r2cx2 zlns
)rd0
W)
8br (alh
+4r
h2
+
2W ,
P)
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where K and E are complete elliptical integrals of first and second kind [12], defined as
E(m)
= j”‘*deJ1_Msin’sdf3.
(B7)
0
0.5
is a number coming from the use of formula (B4) in the range x
b
0
Fig. B2. Plot of the bubble wall tilt angle a versus wall energy variation ratio, derived from eq. (B9).
w
+P(P- Vilfp2).
b is infinite for p = 0 (as seen before) and goes quickly to b = 3/2 + In 2 = 2.1931 when p grows. This limiting value will be used. Deriving with respect to (Y the total energy variation AE, obtained by summing (Bl) and (B6), one gets IaI(ln
Ial-b+c)
Trl h1Vul = --zp u
.
w9
The sign of (Yis such that the head of the cone is in the direction of larger wall energy density u. As usual, the characteristic length I is denoted by 1 = 0/47rM,2, and c is a number depending on 1, h, and r according to 1 +2-x--
+lnk
IT1
16r2 h2
over, this calculation was restricted to a conical shape, but that approximation is thought to be noncritical. From the experiments and from this calculation, it appears that the gradient of magnetization may be the main cause of the tilt. We do not propose here a calculation of this effect, because of the complication of the magnetostatic calculation: the surface charges on the wall are non-constant, and volume charges exist in the domains.
Acknowledgements We thank J. Ben Youssef from L.M.M.M. (Meudon, France) for performing the ferromagnetic resonance measurements at various frequencies, using the strip technique [lo]. J. Miltat contributed many interesting suggestions and comments, which are gratefully acknowledged.
(BlO) References
(c decreases by (h/24r) (H/M,) for an applied field shrinking the bubble). In the case of the standard bubble material, where h = 2r = 10 1, one finds c = 1.1289. Figure B2 plots the solution of (B9) with b and c given above. A tilt angle of 1” is obtained when the relative wall energy increases from 0.15 to 1.85 between the two surfaces. We therefore find that tilts of the order of 1” require quite large variations in wall energy. On the other hand, we did not investigate nonlinear variations of U. More-
[l] See, for example, Magnetic bubbles by A.H. Bobeck and E. Della Torre, in: Selected Topics in Solid State Physics, vol. XIV, ed. E.P. Wolfarth (North-Holland, Amsterdam, 1975). [2] A. Thiaville, L. Arnaud, F. Boileau, G. Sauron and J. Miltat, IEEE Trans. Magn. 24 (1988) 1722. [3] A. Thiaville, J. Ben Youssef, Y. Nakatani and J. Miltat, J.
Appl. Phys. 69 (1991) 6090. [4] In the early days of the bubble memories, published) observed so-called half-bubbles, analyzed by W.J. DeBonte, Bell Syst. Tech. 1933. Bobeck attributed their existence to
Bobeck (unwhich were J. 51 (1972) gradients of
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