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An investigation of magnetic reversal in submicron-scale Co dots
using first order reversal curve diagrams
Chris Pikea)
Department of Geology, University of California, Davis, California 95616
Andres Fernandezb)
Lawrence Livermore National Laboratory, Livermore, California 94551
~Received 2 November 1998; accepted for publication 1 February 1999!
First order reversal curve ~FORC!diagrams are a powerful method of investigating the physical
mechanisms giving rise to hysteresis in magnetic systems. We have acquired FORC diagrams for an
array of submicron-scale Co dots fabricated by interference lithography. These dots reverse
magnetization through the nucleation and annihilation of a single-vortex state. Using FORC
diagrams, we are able to obtain precise values for the nucleation and annihilation fields involved in
magnetic reversal. Our results indicate, however, that there are actually two distinct paths for vortex
annihilation: When a complete magnetic reversal takes place, a vortex enters on one side of a dot
and exits out the opposite side. But if the magnetization is returned to its original orientation before
a complete reversal has occurred, then the vortex will exit on the same side from which it has
entered. We are unable to obtain a precise field value for this later path of annihilation; however, it
is shown that, statistically, the vortex annihilates with greater ease when it exits out the same side
from which it has entered. © 1999 American Institute of Physics. @S0021-8979~99!08409-1#
I. INTRODUCTION
FORC diagrams are a powerful new method of investi-
gating hysteresis in magnetic systems. We have used this
method to study interactions in magnetic recording media1
and multidomain behavior in geological samples.2Here, we
apply the method to an array of polycrystalline Co dots pro-
duced by interference lithography. Small, lithographically
patterned structures such as these have potential applications
in the field of magnetic recording.3,4 This array of Co dots
presents an especially exciting system for a FORC diagram
study, because the dots in this array reverse their magnetiza-
tion through the nucleation and annihilation of a single-
vortex state. With a FORC diagram, we should be able to
obtain precise values for the nucleation and annihilation
fields involved in magnetic reversal. As it turns out, how-
ever, the dynamics of this vortex are more complicated than
expected; our results, described below, indicate that there are
actually two distinct annihilation fields for any given dot.
II. THE SAMPLE
The sample in this study consists of an array of elliptical
Co dots produced by interference lithography and a lift-off
process. A detailed description of the fabrication process is
provided in Ref. 5. A scanning electron micrograph of a
small portion of the array is shown in Fig. 1. The dots have
a 450 nm major axis, a 260 nm minor axis, and a thickness of
30 nm. Cobalt has strong crystalline anisotropy, but the
grains that make up these dots are randomly oriented and
smaller than the theoretical domain wall width. Hence, crys-
talline anisotropy largely cancels, and a uniaxial, in-plane,
shape anisotropy dominates the magnetic behavior of these
samples.5
When the applied field is aligned with the particle easy
~long!axis, this sample has a remanence of Mrs /Ms50.80
and coercivity of 21.2 mT @Fig. 2~a!#. Magnetic force mi-
croscopy ~MFM!investigations have shown that when a
saturating field is aligned with the easy axis and removed,
the dots relax into a uniformly magnetized, single domain
state.5When the field is reversed and increased in the oppo-
site direction, a single-vortex state will nucleate and then
annihilate, leaving a negatively oriented particle. MFM in-
vestigations also show that there is some random variation
from one dot to the next in the value of the nucleation and
annihilation fields.6This variance can be attributed to irregu-
larities in particle shape, size or chemistry. However, the
most likely source of this variance is the random arrange-
ment of grains within each particle.
On the major hysteresis loop, the fields where the slope
has a local maximum provide estimates of the vortex nucle-
ation and annihilation fields. For the applied field aligned
with the particle easy axis @Fig. 2~a!#, we obtain
HN524.0mT and HA588.7mT. However, these estimates
are not very accurate because, in addition to vortex nucle-
ation and annihilation, the reversible movement of the vorti-
ces inside the dots is also contributing to the major hysteresis
loop; there is no way of isolating just the contribution due to
nucleation and annihilation events.
By contrast, since remanence curves are measured at
zero field, vortex movement does not occur, and one is able
to isolate the change in magnetization due to nucleation and
annihilation events. We measured the isothermal remanent
magnetization ~IRM!and dc demagnetization remanence
a!Electronic mail: pike@geology.ucdavis.edu
b!Current address: Etec Systems, Inc., Hayward, CA 94545.
JOURNAL OF APPLIED PHYSICS VOLUME 85, NUMBER 9 1 MAY 1999
66680021-8979/99/85(9)/6668/9/$15.00 © 1999 American Institute of Physics
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~DCD!remanence curves with the applied field aligned with
the particle easy axis ~Fig. 3!.~The IRM remanence curve
begins in a demagnetized state and the DCD curve begins in
a negatively saturated state.!From these curves we estimate
the vortex annihilation field at HA592.6mT. The nucleation
field cannot be estimated from the IRM curve because all the
particles begin in a vortex state, so no nucleation takes place.
The DCD curve begins in a negative saturation state, so
nucleation does take place, but it is difficult to discern where,
in the neighborhood of H50, this curve has a maximum
slope; hence it is difficult to estimate HN.
For the applied field aligned with the hard ~short!axis,
the hysteresis loop has negligible coercivity or remanence
@Fig. 2~b!#. This is because when a saturating field is applied
along the hard axis and removed, the particles relax into a
single-vortex state, as shown by MFM investigations.5From
the major hysteresis loop we estimate the median values:
HN546 mT and HA5169mT. However, as described above,
these estimates have considerable uncertainty. The IRM and
DCD remanence curves are of no use when the applied field
is aligned with the hard axis, since the remanent state is
always a vortex state.
III. FORC DIAGRAMS
A FORC diagram is calculated from a class of hysteresis
curves known as first order reversal curves. As shown in Fig.
4, the measurement of a FORC begins with the saturation of
the sample by a large positive applied field. The field is then
ramped down to a reversal field Ha. The FORC consists of a
measurement of the magnetization as the applied field is in-
creased from Haback up to saturation. The magnetization at
the applied field Hbon the FORC with reversal point Hais
denoted by M(Ha,Hb), where Hb>Ha. A FORC distribu-
tion is defined as the mixed second derivative
FIG. 1. A scanning electron micrograph of a portion of the Co dot array.
The dots, which are produced by interference lithograph and a lift-off pro-
cess, are nominally 450 nm3250 nm in size and 30 nm thick.
FIG. 2. Major hysteresis loop for field aligned with ~a!easy and ~b!hard
particle axis. The values for nucleation and annihilation fields are only
rough estimates.
FIG. 3. IRM and DCD remanence curves for field aligned with easy particle
axis. The annihilation field value is probably a better estimate than that
given by the major hysteresis loop.
FIG. 4. Definition of a first order reversal curve. The measurement of a
FORC begins by saturating the sample with a large positive applied field.
The applied field is then decreased to a reversal field Ha. The FORC is
comprised of a measurement of the magnetization as the applied field is
increased from Haback up to saturation. The magnetization at Hbon a
FORC with reversal field Hais denoted by M(Ha,Hb), where Hb>Ha.
6669J. Appl. Phys., Vol. 85, No. 9, 1 May 1999 C. Pike and A. Fernandez
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r
~Ha,Hb![2
]
2M~Ha,Hb!
]
Ha
]
Hb,~1!
where this is well defined only for Hb.Ha.
When a FORC distribution is plotted, it is convenient to
change coordinates from
$
Ha,Hb
%
to
$
Hc[(Hb2Ha)/2,
Hu[(Ha1Hb)/2}. A FORC diagram is a contour plot of a
FORC distribution with Hcand Huon the horizontal and
vertical axes, respectively. Since Hb.Ha, we have Hc.0,
and a FORC diagram is confined to the right side half plane.
As shown in Ref. 1, for a collection of noninteracting single-
domain particles, Hcis equivalent to particle coercivity.
We have developed a procedure for calculating accurate
FORC diagrams. A detailed description of this procedure can
be found in Ref. 1. The smoothing factor, field spacing, and
averaging time are parameters of the measurement and cal-
culation procedure described in Ref. 1.
IV. EXPERIMENTAL RESULTS
We have investigated the FORC distribution of the Co
dot array and found that it consists of three prominent fea-
tures: two peaks, one in the upper half plane and one in the
lower, and a ‘‘butterfly’’ structure on the central horizontal
axis, which consists, roughly, of a circular region of negative
values superimposed on an elongated horizontal region of
positive values. Rather than displaying the full FORC distri-
bution in one diagram, we have acquired FORC diagrams
which focus in on these specific features. Each of the dia-
grams in this study required the separate measurement of 99
FORCs.
Figures 5–8 were acquired with the applied field aligned
with the particle easy axis. Figure 5 includes both the upper
and lower peaks in order to compare their height. The ratio
of the upper to lower peak was 0.304. Because the lower
peak is so much more pronounced, the upper peak does not
show up well in Fig. 5. Figures 6 and 7 focus on the indi-
vidual peaks. The field spacing for Figs. 6 and 7 was smaller
than that of Fig. 5, giving them greater resolution. The coor-
dinates of the lower and upper peaks were
$
Hc,Hu
%
5
$
49.3,243.360.2mT
%
and $47.8, 44.460.3mT%, respec-
tively. Figure 8 focuses on the ‘‘butterfly’’ structure.
Figures 9–12 were acquired with the applied field
aligned with the hard axis. Figure 9 includes both peaks; the
ratio of upper to lower was 0.45. Figures 10 and 11 focus on
the individual peaks, which have coordinates: $62.0,
2105.360.5mT%and $61.2, 107.260.3mT%. Figure 12 fo-
cuses on the butterfly structure.
V. HYSTERESIS MODELS
To help us interpret these FORC diagrams, we examine
the following four, highly simplified models.
A. Square hysteresis loops
We will initially neglect all reversible changes in mag-
netization, and model the magnetic hysteresis of a particle as
the sum of two square loops, as diagrammed in Fig. 13~a!.In
this diagram, the applied field begins at positive saturation.
When the field falls bellow HN, a vortex nucleates and the
magnetization goes to zero. When the field falls below
2HA, this vortex will exit the particle out the opposite side
from which it has entered; this is necessary to achieve a full
reversal of the magnetization. If the applied field is reversed
after the vortex has nucleated but before it has annihilated
~i.e., before magnetic reversal has been completed!, then
when the field rises above HAthe vortex will exit out the
same side from which it has entered; this is necessary in
order for the magnetization to return to a positive orientation.
Implicit in this model are these two inequalities: 2HA
,HNand HN,HA.
On a FORC curve, the magnetization of the model dia-
grammed in Fig. 13~a!can be written:
M~Ha,Hb!5
u
@Hb2HN1~HN2HA!
u
~HN2Ha!#
2
u
@2HA2Hb1~HA2HN!
u
~2HA2Ha!#,
~2!
where
u
(x)5
$
0 forx,0; 1 forx.0
%
. The upper square
loop in Fig. 13~a!is represented by the first term in Eq. ~2!,
and the lower loop by the second. Taking a derivative with
respect to Hbwe get
dM~Ha,Hb!/dHb5
d
~Hb2HN!
u
~Ha2HN!
1
d
~Hb2HA!
u
~HN2Ha!
1
d
~Hb1HA!
u
~Ha1HA!
1
d
~Hb1HN!
u
~2HA2Ha!.~3!
Taking the derivative with respect to Haand multiplying by
21, we get
FIG. 5. FORC diagram for field aligned with easy axis, displaying both
peaks: field spacing: 1.7 mT, smoothing factor: 5, averaging time: 1 s.
6670 J. Appl. Phys., Vol. 85, No. 9, 1 May 1999 C. Pike and A. Fernandez
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r
~Ha,Hb!52
d
~Hb2HN!
d
~Ha2HN!
1
d
~Hb2HA!
d
~HN2Ha!
2
d
~Hb1HA!
d
~Ha1HA!
1
d
~Hb1HN!
d
~2HA2Ha!.~4!
Since Hb.Ha, the first and third terms equal zero, so
r
~Ha,Hb!5
d
~Hb2HA!
d
~HN2Ha!
1
d
~Hb1HN!
d
~HA1Ha!.~5!
Changing variables to Hcand Hu, we get:
r
~Hc,Hu!5
d
@Hc2~HA2HN!/2#
d
@Hu6~HN1HA!/2#.
~6!
Thus, on a FORC diagram we obtain two point delta func-
tions, one in the upper quadrant and one in the lower, with
coordinates
$
Hc,Hu
%
5
$
(HA2HN)/2,6(HN1HA)/2
%
. The
Hccoordinate is equal to the square loop half width, and the
Hucoordinates are equal to the offset of the square loop
centers. Conversely, the nucleation and annihilation fields
can be calculated from the coordinates of the point delta
functions:
HA5Hu1Hc;HN5Hu2Hc~7a!
using the upper delta point, and
HN52Hc2Hu;HA5Hc2Hu~7b!
using the lower delta point.
Since a real collection of particles will have variations in
chemistry, size, shape and grain structure, these delta func-
tions will be smoothed to some degree.
B. Curvilinear hysteresis loop
Next, we allow a vortex, once it has nucleated, to move
reversibly inside a particle. This will be modeled by letting
the bottom ~top!branch of the upper ~lower!hysteresis loop
be curved as diagrammed in Fig. 13~b!. On a FORC curve,
the magnetization on the upper loop is
Mupper~Ha,Hb!
5
u
@Hb2HN1~HN-HA!
u
~HN2Ha!#1f~Hb!
3
$
12
u
@Hb2HN1~HN2HA!
u
~HN2Ha!#
%
~8!
where f(H) is the magnetization on the bottom branch of the
upper loop and on the top branch of the lower loop. Evalu-
ating the mixed derivative and multiplying by 21, we get
r
upper~Ha,Hb!5@12f~HA!#
d
~Hb2HA!
d
~HN2Ha!
1@df~Hb/dHb!#
d
~HN2Ha!
u
~Hb2HN!
3
u
~2Hb1HA!2@df~Hb!/dHb#
3
d
~HN2Ha!
u
~HN2Hb!
u
~Hb2HA!.~9!
Since HN,HA, the third term in Eq. ~9!is zero. The first
term yields the same upper point delta function which we
obtained in the previous section, but with a prefactor @1
2f(HA)#. The second term generates positive values at Ha
FIG. 6. Easy axis. Lower peak at $48.7, 243.360.2 mT%: field spacing: 1.1
mT, smoothing factor: 5, averaging time: 1 s.
FIG. 7. Easy axis. Upper peak at $47.8, 44.260.3 mT%: field spacing: 0.8
mT, smoothing factor: 7, averaging time: 1 s.
FIG. 8. Easy axis. Butterfly structure: field spacing: 0.9 mT, smoothing
factor: 9, averaging time: 1 s.
6671J. Appl. Phys., Vol. 85, No. 9, 1 May 1999 C. Pike and A. Fernandez
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5HN, for HN,Hb,HA. On a FORC diagram, this is a
145° line between the upper point delta function and the
Hc50 axis.
The magnetization on the lower loop in Fig. 13~b!is
Mlower~Ha,Hb!
52
u
@2HA2Hb1~HA2HN!
u
~2HA2Ha!#
2f~Hb!
$
21
1
u
@2HA2Hb1~HA2HN!
u
~2HA2Ha!#
%
.~10!
Evaluating the mixed derivative we get
r
lower~Ha,Hb!5@11f~2HN!#
d
~Hb1HN!
d
~Ha1HA!
1@df~Hb!/dHb#
d
~Ha1HA!
u
~2HA2Hb!
3
u
~HN1Hb!2@df~Hb!/dHb#
3
d
~Ha1HA!
u
~HA1Hb!
u
~2HN2Hb!.
~11!
Since HN,HA, the second term in Eq. ~11!is equal to zero.
The first term yields the same lower point delta function
which we obtained in Sec. VA, but with a prefactor @1
1f(2HN)#. The third term generates negative values at
FIG. 9. Field aligned with hard axis. Both peaks: field spacing: 3.4 mT,
smoothing factor: 4, averaging time: 0.5 s.
FIG. 10. Hard axis. Lower peak at $61.5, 2106.060.5 mT%: field spacing:
1.7 mT, smoothing factor: 5, averaging time: 0.4 s.
FIG. 11. Hard axis. Upper peak at $60.2, 107.260.3 mT%: field spacing: 1.7
mT, smoothing factor: 7, averaging time: 0.6 s.
FIG. 12. Hard axis. Butterfly structure: field spacing: 1.7 mT, smoothing
factor: 10, averaging time: 0.6 s.
6672 J. Appl. Phys., Vol. 85, No. 9, 1 May 1999 C. Pike and A. Fernandez
Downloaded 04 Dec 2001 to 169.237.43.193. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
Ha52HA, for 2HA,Hb,2HN. On a FORC diagram,
this is a 145° line between the lower point delta function
and the Hc50 axis.
In summary, because of reversible vortex movement, the
ratio of the magnitude of the upper delta point function to the
magnitude of the lower delta function is @12f(HA)#/@1
1f(2HN)#. It can be seen from Fig. 13~b!that this is less
than one; so the lower peak has a greater magnitude than the
upper peak. Reversible movement also results in positive
~negative!values on a 145° line between the upper ~lower!
point delta function peak and the Hc50 axis.
C. Two uncorrelated annihilation fields
It can be expected that particle irregularities such as
grain texture will play a role in the nucleation and annihila-
tion of vortices. Hence, there will be one site on a particle
where vortex nucleation is favored. As mentioned earlier,
there are actually two paths for vortex annihilation: a vortex
can exit a dot out the same side from which it has entered, or
out the opposite side. Due to irregularities, these two differ-
ent paths will have two different annihilation fields for an
individual particle. We therefore generalize our model to in-
clude two distinct annihilation fields, as diagrammed in Fig.
13~c!. We define HA1as the annihilation field when annihi-
lation takes place on the opposite side of nucleation, and HA2
as the annihilation field when annihilation takes place on the
same side. Implicit in this model are the inequalities: HN
,
$
HA1,HA2
%
and HN.
$
2HA1,2HA2
%
.
On a FORC the hysteresis behavior diagrammed in Fig.
13~c!can be expressed as:
M~Ha,Hb!
5
u
@Hb2HN1~HN2HA2!
u
~HN2Ha!
1~HA22HA1!
u
~2HA12Ha!#
2
u
@2HA12Hb1~HA12HN!
u
~2HA12Ha!#.~12!
In Appendix A, the mixed second derivative of Eq. ~12!is
evaluated. The resulting expression is
r
~Ha,Hb,HN,HA1,HA2!
5
d
~Hb2HA2!
d
~HN2Ha!
u
~HA11Ha!
2
d
~Hb2HA2!
u
~HN2Ha!
d
~HA11Ha!
1
d
~Hb2HA1!
d
~2HA12Ha!
1
d
~Hb1HN!
d
~2HA12Ha!.~13!
Next, let us consider a collection of particles with a sta-
tistical distribution of nucleation and annihilation fields.
Here, we will treat the two annihilation fields, HA1and HA2,
as independent variables which have the same statistical dis-
tribution. That is, the values of HA1and HA2for a given
particle will not be correlated, but over the collection of par-
ticles they will have the same statistical distribution. To sim-
plify the analysis it is advantageous to write the distribution
of particles as the product of three distributions:
r
N(HN)
r
A(HA1)
r
A(HA2), where
r
Ndenotes the distribution
of nucleation fields and
r
Adenotes the distribution of anni-
hilation fields. This is not generally possible because of the
inequalities HN,
$
HA1,HA2
%
and HN.
$
2HA1,2HA2
%
.
However, if it is assumed that
r
N(x) is distributed at smaller
values than
r
A(x), with no overlap, then the first constraint
will always be satisfied. Similarly, if it is assumed that
r
N(x)
is distributed at larger values than
r
A(2x), again with no
overlap, then the second constraint will always be satisfied.
We can then write the distribution of particles as the above
described product.
r
(Ha,Hb) is obtained by integrating
r
(Ha,Hb,HN,HA1,HA2) in Eq. ~13!over these distribu-
tions. This integration is done in Appendix A. The result is
r
~Ha,Hb!5
EEE
2`,`dHNdHA1dHA2
r
N~HN!
r
A~HA1!
3
r
A~HA2!
r
~Ha,Hb,HN,HA1,HA2!~14!
5
r
N~Ha!
r
A~Hb!2
r
A~2Ha!
r
A~Hb!
1
r
A~Hb!
d
~Ha1Hb!1
r
N~2Hb!
r
A~2Ha!.
~15!
The first and last terms generate peaks in the upper and
lower quadrants of a FORC diagram. The two middle terms,
however, give rise to a new feature: a butterfly structure on
the central horizontal axis. This butterfly structure is a con-
sequence of the presence of two uncorrelated annihilation
fields. To illustrate the general features of the model, we
have changed variables and calculated a FORC diagram of
r
(Hc,Hu) using Gaussian functions for
r
Nand
r
A, with
means of 0.2 and 2.0, respectively, and standard deviations
of 0.43 ~Fig. 14!.
FIG. 13. Particle magnetization modeled by ~a!two square loops, ~b!two
curvilinear loops, ~c!two square loops with different annihilation fields.
6673J. Appl. Phys., Vol. 85, No. 9, 1 May 1999 C. Pike and A. Fernandez
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D. Two annihilation fields;
H
A
2distributed at smaller
values
In Sec. VC, it was assumed that the two annihilation
field values, HA1and HA2, had the same statistical distribu-
tion. But in actuality, a distinction between these two distri-
butions is expected for the following reason: When a vortex
nucleates, it will pick that site on the particle surface where
irregularities cause the greatest reduction in the nucleation
energy barrier. It is reasonable to assume that the same ir-
regularities which lower the energy barrier to vortex nucle-
ation will also lower the energy barrier to vortex annihila-
tion. Therefore, the location on the particle where vortex
nucleation takes place will also tend to be a favorable loca-
tion for vortex annihilation. Since HA2is defined as the an-
nihilation field when a vortex exits on the same side it enters,
then it follows that HA2will tend to be smaller than HA1.
To model this effect, we shift the distribution of HA2to
smaller values. We will write the distribution of HA1as
r
A1(HA1) and the distribution of HA2as
r
A1(HA21k) where
kis positive. Then Eq. ~14!becomes
r
~Ha,Hb!5
EEE
2`,`dHNdHA1dHA2
3
r
N~HN!
r
A1~HA1!
r
A1~HA21k!
3
r
~Ha,Hb,HN,HA1,HA2!.~16!
This integration is evaluated in Appendix A. We get
r
~Ha,Hb!
5
r
N~Ha!
r
A1~Hb1k!
2
r
A1~2Ha!
r
A1~Hb1k!1
r
A1~Hb!
d
~Ha1Hb!
1
r
N~2Hb!
r
A1~2Ha!.~17!
The last term in Eq. ~17!generates a peak in the lower
quadrant of a FORC diagram. This peak will be located at
Ha52
^
HA1
&
and Hb52
^
HN
&
, where
^
HN
&
and
^
HA1
&
de-
note the medians of the distributions of HNand HA1, respec-
tively. On a FORC diagram, after changing variables to Hc
and Hu, this lower peak will have coordinates
$
Hc,Hu
%
5
$
(
^
HA1
&
2
^
HN
&
)/2,2(
^
HA1
&
1
^
HN
&
)/2
%
. Conversely, we
can calculate
^
HN
&
and
^
HA1
&
from the coordinates of the
lower peak:
^
HN
&
52Hc2Hu;
^
HA1
&
5Hc2Hu~18a!
The first term in Eq. ~17!generates a peak in the upper
quadrant. This peak will be located at Ha5
^
HN
&
and Hb
5
^
HA1
&
2k. Since
r
A1(HA21k) is the distribution of the
annihilation field HA2, we can write
^
HA2
&
1k5
^
HA1
&
,so
H
b
5
^
H
A2
&
. On a FORC diagram, after changing variables
to Hcand Hu, this upper peak will have coordinates
$(
^
HA2
&
2
^
HN
&
)/2, (
^
HA2
&
1
^
HN
&
)/2%. Conversely, we can
calculate
^
HN
&
and
^
HA2
&
from the coordinates of the upper
peak:
^
HN
&
5Hu2Hc;
^
HA2
&
5Hu1Hc.~18b!
To illustrate the features of this model, we changed variables
and calculated a FORC diagram of
r
(Hc,Hu) for k50.09
using the distribution functions of Sec. V. C and letting
r
A15
r
A. As seen in Fig. 15, the negative region of the
butterfly structure has been shifted leftward and downward
relative to the positive region.
VI. COMPARISON OF THEORY AND EXPERIMENT
On the basis of the above-described models, we can
identify the peaks in our FORC diagrams as manifestations
of the nucleation and annihilation of single-vortex states. The
presence of the butterfly structures in Figs. 8 and 12 is con-
sistent with the model of Sec. VC and is an indication that
there are two distinct annihilation processes. The region of
positive values contained in this butterfly structure is shifted
to the left and downward relative to the region of negative
values. This is most evident when the applied field is aligned
with the particle easy axis ~Fig. 8!. This shift is consistent
FIG. 14. FORC diagram for two uncorrelated annihilation fields having
identical distributions. Calculated using Eq. ~15!.
r
Nand
r
Ahave means 0.2
and 2.0, respectively, and standard derivations of 0.42. FIG. 15. FORC diagram calculated using Eq. ~17!with k50.09, where
r
N
and
r
A1have means 0.2 and 2.0, respectively, and standard derivations of
0.42, and where the distribution of HA2is given by
r
A1(HA21k).
6674 J. Appl. Phys., Vol. 85, No. 9, 1 May 1999 C. Pike and A. Fernandez
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with the results of the model in Sec. VD and indicates that
the median value of HA2is less than that of HA1. In other
words, a vortex is annihilated with greater ease when it exits
out the same side from which it has entered.
We have also observed a region of positive values on a
145° line between the upper peak and the Hc50 axis ~Figs.
7 and 11!, and a region of negative values on a 145° line
between the lower peak and the Hc50~Figs. 6 and 10!.
These features are consistent with the model in Sec. VB and
can be identified as manifestations of the reversible move-
ment of the vortices. In addition, it has been found that the
upper peak has a smaller magnitude than the lower peak
~Figs. 5 and 9!. In Sec. VB, this was also shown to be a
manifestation of reversible vortex movement. In the case of
the particular sample studied here, the reversible movement
of the vortex is large enough that it has reduced the magni-
tude and clarity of the upper peaks, particularly when the
field is aligned with the easy axis ~Fig. 5!. It appears that
there is greater reversible movement when the applied field
is aligned with the easy axis.
Using Eq. ~18b!, the median value of the vortex nucle-
ation field can be calculated from the coordinates of the up-
per peak. For the applied field aligned with the particle easy
axis, and with the coordinates of the upper peak from Fig. 7,
we get
^
HN
&
523.660.5mT. Using Eq. ~18a!, the median
annihilation field HA1can be calculated from the coordinates
of the lower peak. With the coordinates of the lower peak
from Fig. 6, we get
^
HA1
&
592.060.4mT.
For the applied field aligned with the hard axis, and us-
ing the peaks in Figs. 10 and 11, the same calculation gives
us
^
HN
&
547.060.8mT and
^
HA1
&
5167.560.8mT. Hence,
when the applied field is aligned with the hard axis, the vor-
tex nucleates with greater ease, and annihilates with greater
difficulty. The different nucleation and annihilation fields as-
sociated with the two orientations can be qualitatively ex-
plained as being due to the different demagnetization fields.7
In the model of Sec. VD, expressions were also derived
to calculate
^
HN
&
from the lower peak and
^
HA2
&
from the
upper peak. Unfortunately, the reversible movement of the
vortex invalidates these calculations. For example, the anni-
hilation event at HA2in Fig. 13~c!will be preceded by a
reversible increase in magnetization. On a FORC diagram,
this reversible increase generates positive values on a 145°
line from the Hc50 axis to the upper peak. The upper peak
will be superimposed upon this 145° line. This causes an
apparent shift in the peak’s location downward and to the left
along a 145° line, which skews an estimate of
^
HA2
&
to-
wards reduced values.
The peaks in these FORC diagrams have considerable
spread. This indicates that there is a variance in the nucle-
ation and annihilation field values among the collection of
particles. We attribute this variance to the random arrange-
ment of grains within a particle. We can quantify the vari-
ance of the nucleation field HNby taking the 1/2 width of a
cross section at a 245° angle through the upper peak on a
FORC diagram. For the field aligned with the easy axis, we
calculated the 1/2 width of the HNdistribution to be 8.4 mT.
MFM measurements of a similar dot array have shown that
there is a variance in nucleation fields of 7.5 mT. This cor-
responds to a 1/2 width of 8.8 mT, in excellent agreement
with the value obtained here.6
@There is a question of how much of the spread on
a FORC diagram is real and how much is due to nu-
merical effects. It can be shown that a feature of a FORC
diagram which would have a length equal to
~112*smoothing factor!*~field spacing!in the absence of
numerical effects, will be increased in length by 8% on a
FORC diagram calculated numerically from data; larger fea-
tures will be increased by smaller fractions. In the numerical
calculation just described, field spacing50.8 mT, smoothing
factor52, and ~112*smoothing factor!*~field spacing!54
mT, which is considerably smaller than the calculated peak
1/2 width of 8.4 mT. Therefore, we can conclude that this
spread is, to a good approximation, due to variance in the
underlying particle nucleation fields.#
VII. CONCLUSIONS
These FORC diagrams indicate that there are actually
two distinct paths of vortex annihilation: the vortex can exit
out the opposite side of the particle from which it has en-
tered, or it can exit out the same side. The former completes
a particle’s magnetic reversal, the latter returns a particle to
its original magnetic orientation. Within an individual par-
ticle, these two paths can have different annihilation field
values; we attribute this to the effect of grain texture. Using
a FORC diagram, we have obtained precise median values
for the nucleation field and for the annihilation field ~i.e.,
HA1!associated with a magnetic reversal. These values are
much more precise than can be obtained via the major hys-
teresis loop or remanence curves. However, we were unable
to obtain a precise value for the annihilation field associated
with returning a particle to its original orientation ~i.e., HA2!;
it was not possible to decouple this annihilation event from
the reversible movement of the vortices. We were able to
show, however, that the HA2is statistically smaller than
HA1. In other words, annihilation tends to occur more easily
when the vortex exists out the same side from which it has
entered. This indicates that the same grain texture which pro-
motes the nucleation of a vortex also promotes its annihila-
tion.
ACKNOWLEDGMENTS
The authors are grateful to Dr. Kenneth Verosub of the
Geology department at U. C. Davis for giving them use of
his facilities.
APPENDIX A
Taking the derivative of Eq. ~12!with respect to Hb,
dM~Ha,Hb!/dHb5
d
~Hb2HN!
u
~2HN1Ha!
u
~HA11Ha!
1
d
~Hb2HA2!
u
~HN2Ha!
u
~HA11Ha!
1
d
~Hb2HA1!
u
~HN2Ha!
u
~2HA12Ha!
1
d
~Hb1HA1!
u
~HA11Ha!
1
d
~Hb1HN!
u
~2HA12Ha!.~A1!
6675J. Appl. Phys., Vol. 85, No. 9, 1 May 1999 C. Pike and A. Fernandez
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Since 2HA1,HN, then
u
(2HN1Ha)
u
(HA11Ha)5
u
(Ha
2HN) and
u
(HN2Ha)
u
(2HA12Ha)5
u
(2HA12Ha).
Taking the derivative with respect to Haand multiplying by
21weget
r
~
H
a,H
b,H
N
H
A1,H
A2
!
52
d
~Hb2HN!
d
~Ha2HN!
1
d
~Hb2HA2!
d
~HN2Ha!
u
~HA11Ha!
2
d
~Hb2HA2!
u
~HN2Ha!
d
~HA11Ha!
1
d
~Hb2HA1!
d
~2HA12Ha!
2
d
~Hb1HA1!
d
~Ha1HA1!
1
d
~Hb1HN!
d
~2HA12Ha!.~A2!
Since Hb.Ha, then the first and fifth terms are zero. After
omitting these terms, we get Eq. ~13!.
Next, to evaluate Eq. ~14!we integrate
r
(Ha,Hb,HN,HA1,HA2) in Eq. ~13!over the distributions
of HN,HA1and HA2. We will perform this integration on
the four terms of Eq. ~13!one at a time:
~first term!
E
2`
`dHNdHA1dHA2
r
N~HN!
r
A~HA1!
r
A~HA2!
3
d
~Hb2HA2!
d
~HN2Ha!
u
~HA11Ha!
5
r
N~Ha!
r
A~Hb!
E
2`
`dHA1
r
A~HA1!
u
~HA12Ha!
5
r
N~Ha!
r
A~Hb!.
The last step uses the fact that, since
r
N(x) is distributed at
smaller values than
r
A(x) with no overlap, then
r
N(Ha)
r
A(HA1) will equal zero whenever HA1,Ha, and
therefore the step function can be equated with 1.
~second term!
2
EEE
2`,`dHN...
r
A~HA2!
3
d
~Hb2HA2!
u
~HN2Ha!
d
~HA11Ha!
52
r
A~2Ha!
r
A~Hb!
E
2`
`dHN
r
N~HN!
u
~HN2Ha!
52
r
A~2Ha!
r
A~Hb!.
The last step uses the fact that, since
r
N(x) is distributed at
larger values than
r
A(2x) with no overlap, then
r
A(2Ha)
r
N(HN) will equal zero whenever HN,Ha, and
the step function can be equated with 1.
~third term!
EEE
2`,`dHN...
r
A~HA2!
d
~Hb2HA1!
d
~2HA12Ha!
5
r
A~Hb!
d
~Ha1Hb!.
~fourth term!
EEE
2`,`dHN...
r
A~HA2!
d
~Hb1HN!
d
~2HA12Ha!
5
r
N~2Hb!
r
A~2Ha!.
Summing these four terms, we get Eq. ~15!.
Next, to evaluate Eq. ~16!we replace
r
A(HA1) with
r
A1(HA1) and
r
A(HA2) with
r
A1(HA21k). Following the
same steps taken above, we get:
~first term!
r
N(Ha)
r
A1(Hb1k).
~second term!2
r
N(2Ha)
r
A1(Hb1k).
~third term!
r
A1(Hb)
d
(Hb1Ha).
~fourth term!
r
N(2Hb)
r
A1(2Ha).
Summing these four terms gives us Eq. ~17!.
1C. R. Pike, A. P. Roberts, and K. L. Verosub, J. Appl. Phys. 85, 6660
~1999!.
2C. R. Pike, A. P. Roberts, and K. L. Verosub, J. Geophys. Res. ~submit-
ted!.
3L. Kong, L. Zhuang, and S. Y. Chou, IEEE Trans. Magn. MAG-33, 3019
~1997!.
4R. M. New, R. F. W. Pease, and R. L. White, J. Magn. Magn. Mater. 155,
140 ~1996!.
5A. Fernandez, M. R. Gibbons, M. A. Wall, and C. J. Cerjan, J. Magn.
Magn. Mater. 190,71~1998!.
6A. Fernandez and C. J. Cerjan, J. Appl. Phys. ~submitted!.
7W. Wernsdorpher et al., Phys. Rev. B 53, 3341 ~1996!.
6676 J. Appl. Phys., Vol. 85, No. 9, 1 May 1999 C. Pike and A. Fernandez
Downloaded 04 Dec 2001 to 169.237.43.193. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp
