CC BY
9Scientific Notes of Taurida National V. I. Vemadsky University
Series : Physics and Mathematics Sciences. Volume 26 (65). 2013. No. 2. P. 38-52
UDK537.6, 538.9
DISTRIBUTION OF THE ANTIFERROMAGNETIC VECTOR FOR A PERIODIC SYSTEM OF REMOTE CIRCULAR ANTIDOTS AND A COUPLE OF CIRCULAR ANTIDOTS IN AN ANTIFERROMAGNETIC FILM
Gorobets Yu. I.1, Gorobets O. Yu.2, Kulish V. V.2
1Institute of Magnetism, National Academy of Sciences of Ukraine, 36-b Vernadsky Blvd, Kyiv
03142, Ukraine
2
National Technical University of Ukraine "Kyiv Polytechnic Institute", 37 Peremogy Prospect, Kyiv
03056, Ukraine
E-mail: kulish [email protected]
The paper is dedicated to the theoretical investigation of the antiferromagnetic vector distribution in a film composed of an isotropic or uniaxial antiferromagnet in the presence of circular antidots. A solution of the Landau-Lifshitz equation is obtained for the antiferromagnetic vector in such antidot system. The antiferromagnetic vector distribution is obtained for a periodic system of remote antidots in an isotropic antiferromagnet and for a couple of antidots (not remote, in general) in an isotropic antiferromagnet, "easy axis" antiferromagnet and "easy plane" antiferromagnet with various boundary conditions. Keywords: antiferromagnet, magnetic thin film, magnetic antidot, antiferromagnetic vector.
PACS: 75.50.Ee, 41.20.Gz, 75.70.Ak
INTRODUCTION
It is known that magnetic nanostructures of different configurations - magnetic quantum dots [1], magnetic thin films [2], magnetic nanospheres [3], nanowires [4] and nanotubes [5], magnetophotonic crystals [6] and other nanostructures - are investigated intensively in recent years. These nanostructures have found a multitude of technical applications - in information storage and transmission devices [7], in magnetic resonance imaging [8], for magnetic refrigeration [9] and so on. In particular, magnetic quantum dots and their systems as well as magnetic antidots [10] (holes in thin films) and their systems are perspective for various technical applications.
Systems of ferromagnetic [11, 12] and antiferromagnetic [13, 14] dots are studied intensively in recent years. However, systems of ferromagnetic [15, 16] and especially antiferromagnetic [17] antidots are poorly researched at the moment, and known studies in this area are devoted mainly to exchange bias in antiferromagnetic antidot arrays. However, antiferromagnetic antidot systems are promising for a variety of technical applications - in information storage devices [18], in magnon waveguides [19], as a basis for magnetic metamaterials [20], as two-dimensional magnonic crystals [21] and so on. Thus, magnetic properties of antiferromagnetic antidots and their systems are an actual topic of research.
We consider a system of circular antidots in an antiferromagnetic film. We obtain a distribution of the antiferromagnetic vector for the following antidot systems (with various boundary conditions): a couple of circular antidots in an isotropic antiferromagnet, a couple of circular antidots in an uniaxial antiferromagnet and a periodic remote antidot system in an isotropic antiferromagnet.
38
1. SETTING OF THE PROBLEM
Let us consider a two-sublattice antiferromagnetic film with a thickness d and direct Oz axis in the normal to the film direction. Let us also consider a system of circular antidots in this film, with radiuses R' and in-plane radius-vectors of the antidots axes {r0i} (see Fig. 1). We assume that the magnetization density of the antiferromagnet sublattices (M1 and M2, respectively) are equal in magnitude and opposite in direction, so that M1 = - M2, and are constant in magnitude, so that |M1| = |M2| = M0, M0 = const, everywhere in the film. Thus, the total magnetization vector M = 0, and the antiferromagnetic vector is also constant in magnitude: |Z,| = L0 = const. We also assume that the film antiferromagnet is characterized by the following parameters: uniaxial anisotropy constants fa and non-uniform exchange constants aj and a2 (where aj > 0), a uniform exchange constant A.
The goal of this work is to find a distribution of the antiferromagnetic vector in the above-described antiferromagnetic film for a periodic remote antidot system and a couple of antidots (in general, not a remote couple) with various boundary conditions.
Fig. 1. Antiferromagnetic film investigated in the paper.
2. SOLUTION OF THE LANDAU-LIFSHITZ EQUATION. GENERAL FORM OF THE ANTIFERROMAGNETIC VECTOR DISTRIBUTION
Let us write down the Landau-Lifshitz equation for a static distribution of the antiferromagnetic vector L in the antiferromagnetic film that we consider in spherical coordinates (r,6,q>). If we denote the azimuthal and polar angles of the vector L as yL and 6L, respectively, the equation can be written in the following form [22, 23]:
c2d/'v(sin2 0LV^L )= 0
2A0l + ((gH0)2 - Cq A^l - ®02 )sin eL cos 0L = 0,
_ 4^0M0 r-— M I-
here H0 is the external magnetic field, c0 = ~ yJA&1 , o0 =—0—Afix .
n h
In the absence of the external field (H0 = 0) the equation (1) - after dividing on the value a)0 - becomes similar to the static equation for the magnetization in an uniaxial
39
ferromagnet with the exchange constant a1 and the anisotropy constant This allows us to use the solutions of this equation are given in [23]. Let us select the following solution:
f^ V , b ,
2 J c/n(cJ\C\P(X, Y, Z), k, ) , (2)
çL = Q(X, Y, Z )
here X = x / /0, Y = y / /0, Z = z / /0; value /0 /|^ | for fa ^ 0, l0 = 1 for fa = 0;
C1 is a constant that lies in the range - 1/4 < C1 < 0 for this solution, and the values
c =
1 + 2Cj + ^ 1 + 4Cj
2 C*l
b =
1 + 2Cj - 41 + 4C1
2 Cj
, ¿1 =
2^1 + 4C1
1 + 2C1 + 7 1 + 4C1 '
(3)
0 < < 1 is the modulus of the Jacobi elliptic function. The functions P and Q for the antidot system described in the "Setting of the problem" section can be written down in the following form:
p = ^ ©a )+z n in
l0 i ± z
r - r
1Л
V l0
+ C,
(4)
Q = y- ©(-A )+Za
i
,n, + C
here r is the radius vector of a point in a plane, r0i =(x0i,y0i) are in-plane radius vectors of the antidots centers, ni are arbitrary integers, C2, C3 are constants, ai is the azimuthal angle
relative to the point r0i (so that a i = arctg
f У - У оЛ
) and the function
V x x0i J
fo, о 11, £> 0
(5)
Let us consider specific antidots configurations and find forms of the solution (4) for these configurations with specific boundary conditions.
3. COUPLE OF ANTIDOTS IN AN ISOTROPIC ANTIFERROMAGNET
Let us select the functions P and Q in the following form:
r
P = n, ln — + n ln 1 l
i\r - r ^ ' о
V
l
+ C
0 J
(6)
Q = nlç + n2a + C3
<
40
where a is an azimuthal angle relative to the point r0. Such distribution corresponds, in particular, to a couple of circular antidots in an isotropic antiferromagnet with their centers in the origin of coordinates and in the point r0. We consider the couple of antidots of the same radii R; for the sake of convenience we choose the coordinate system so that the axis
I d \
of the antidot 2 lies on the semiaxis x > 0, so that r =
V 0 y
d > 0.
Let us consider the case when these antidots are not remote from one another, so that, in general, the condition |r0| » R is not satisfied. Then, the functions P and Q have the following form on the antidots surfaces:
Q = n1® + n2
on the surface of the antidot 1, and
P = n ln R + n2 ln
1 l
I r\
f
d
R2
R
f
V
1H—--2—cos® d2 d
+ C
(7)
t - arctg
sin®
d/R - cos® j
+ C
f
P = n ln
d L R2 „R
— J1H—- + 2—cos a L\ d2 d
\
+ n2 ln
J
fR}
V l0 J
+ C
(8)
Q = niarctg I
sin a
d / R + cos a
+ n a + C,
on the surface of the antidot 2.
As we can see from (7) and (8), a simple and convenient choice of boundary conditions - constant boundary conditions on antidots surfaces - cannot be made in this case because the functions P and Q depend on the local azimuthal angle on the surface of each antidot. However, we can set boundary conditions in some points of the antidots surface, for example, on the intervals y = 0 (for the first antidot) or a = 0 (for the second antidot). In addition, we can set boundary conditions on some other interval of the
r d /2 ^
antiferromagnetic film, for example, in the point
0
. Let us consider various choices
J
of the boundary conditions.
Boundary conditions 1.
On the surface of the antidot 2 6L = n/2, yL = y0 for a = 0, on the surface of the antidot 1 6L = n/2 for y = 0.
Substituting these boundary conditions into the solution (2), (6), we obtain
l
0
41
tg
(6
dn
'Vl^il
( f n ln
r - e d
\
L
+ n ln
f \ ^ ^ r
+ C
V 0 Q = n1ç + n2a +
V lo J
k
J J
(9)
where ex is an ort of the Ox axis and the constants Ci, C2 should be found from the conditions
( , (d + n1 ln
+ n2 ln
/
V lo J R
f R} l
+ C
= F
f
n ln —h n2 ln 1 l
V
d - R
"T"
V'0 J J
A
arcsin
+ C
(
= F
V '0 J
arcsin
kl
VT-b2 k
A
, k
+ 4 K (kT )N
, (10)
,k
+ 4K (kl N 2
here ^i and A2 are integers and ^ (£ £) is the incomplete elliptic integral of the first kind:
s
F (S, k ) = J
dp
VT—k
2 • 2 ' Sin p
(11)
Systems (9) and (10) determine the sought constants C1, C2.
Boundary conditions 2.
On the surface of the antidot 2 6L = n/2, yL = y0 for a = 0, 6L = n/2 in the point
( d /2 ^ v 0 J
In this case, the solution (2), (6) takes an analogous to the previous case form (9). Let us find the distribution constants C1, C2. From the boundary conditions on the surface of the antidot 2 we obtain
id + R\ i n\ \ ( h jJ ^
I n1ln
V l0 J
+ n2 ln
f R} V l0 J
+ C
= F
. 7
arcsin-, k
k 71
+ 4K(k1 )Nj, (12)
where N1 is an arbitrary integer. Distribution (6) in the point
d
( d /2 ^ v 0 J
has the following form:
P = (n + n2 )ln--h C2
v 1 2/ 2L 2
(13)
Q = n2n + C3
hence, using the boundary condition in the point
( d /2 ^ v 0 J
we obtain
b
<
<
42
I (n + n )ln
d_ 2L
+ C
= F
arcsin -
. Vl - b7
k
-, ki
+
4^ (ki )N 2
(14)
where N2 is an arbitrary integer. The system (12), (14) determines the constants C1 and C2.
Boundary conditions 3.
On the surface of the antidot 2 6L = n/2 for a = 0, yL = y0 and 6L = n/2 in the point
r d /2 ^
V 0 y
From the boundary condition on the surface of the antidot 2 we obtain
^ r VTb2 ^
yl\c1\ I niln
C d + R^
V lo y
+ n2 ln
R
V lo y
+ C
= F
arcsin -
ki
+ 4^ (ki )Ni, (15)
where N1 is an arbitrary integer. After taking into account (13), from the boundary
r d /2 ^
condition in the point
V0y
implies
JcJ I (n + n2 )ln ~~~ + C
(
2L
= F
. yfl-b1 . arcsin-, k
\
+ 4^ (ki )N 2
(16)
C3 =^0 - n2^
where is an arbitrary integer. The system (15), (16 ) determines the constants Cu C2 and C3.
4. COUPLE OF ANTIDOTS IN AN "EASY PLANE" ANTIFERROMAGNET
Let us select the functions P and Q in the following form:
1
P = ni ln — + n2 ln i l
z
f
1 -1
W
l
+ C
V y
(i7)
Q = —h + n2a + C3 l
where a is an azimuthal angle relative to the point r0. Such distribution corresponds, in particular, to a couple of circular antidots in an uniaxial antiferromagnet with the "easy plane" anisotropy. Similarly to the previous case we assume that antidots, in general, are not remote and that their radii R are the same, and we choose the coordinate system in a
r d ^
, d > 0. Thus, the functions P and Q on the antidots surfaces
similar way, so that r0 = have the following form:
V 0 y
2
43
Q = J- + niV + n2
on the surface of the antidot 1, and
P = n1 ln — + n2 ln 1 l
f d I R2 R ^ ^ — J1H—--2 — cos® + C L\ d2 d
f
n - arctg
sin®
A
(18)
f
P = n ln
d L —2 nR
—, 1H—- + 2 — cos a ln\ d2 d
d / R - cos®
"A
+ n2 ln
y
V l0 y
+ C,
+ C2
(19)
Q = — + narctg I
V
sin a
d / R + cos a
+ n a + C
on the surface of the antidot 2.
As we can see, because of the dependence of the function Q on z we cannot set
constant boundary conditions for yL on some interval
ir }
'1
® y
. However, we can set
boundary conditions in some point
fr\ ®1
V z1 y
and for the sake of simplicity we can choose the
origin of the Oz axis in the plane z = zh Let us illustrate this approach on three different sets of boundary conditions, analogous to the previous (antidots couple in an isotropic antiferromagnet) case.
Boundary conditions 1.
On the surface of the antidot 2 6L = n/2 for a = 0, yL = y0 for a = 0, z = 0; on the surface of the antidot 1 6L = n/2 for y = 0.
From the boundary condition for yL we obtain C3 = y0, so the solution (2), (17) in this case can be rewritten
tg
(6
b
dn
( f n1 ln
r - exd l
\
+ n2 ln
0 y
fL\ V l0 y
+ C
*1
yy
(20)
Q = —H n1® + n2a + ®0 l
with constants Ci, C2 determined from the conditions similar to the previous case:
44
'VlC1 I
( , r d+^
n1 ln
+ n2 ln
/
V 40 y
R
n ln--h n ln
1 l
V
V l0 y
^d - R
+ C
r
= F y
A r
arcsin
V lo y
+ C
= F
arcsin
VT-b k1
VT-b2 k
, Ik,
, Ik,
+ 4* (k )N + 4 * (kl )N2
, (21)
here N1, N2 are arbitrary integers. Systems (20) and (21) determine the sought solution for the boundary conditions we consider.
Boundary conditions 2.
On the surface of the antidot 2 6L = n/2 for a = 0, (L = (0 for a = 0, z = 0; 6L = n/2 in
r d /2 ^
the point .
10 y
In an analogous way to the previous boundary condition choice we obtain C3 = (0, so the solution (2), (17) takes the form (20). Distribution constants C1, C2 are found analogously to the previous case, boundary conditions 2: from the boundary conditions on the surface of antidot 2 we obtain
fd + r»\ \ r R ~îJ \
ylKlI n1ln
V l0 y
+ n2 ln
R
V l0 y
+ C,
= F
arcsin -
vr-b
, k
+ 4* (k )N1, (22)
where N1 is an arbitrary integer, and from the boundary conditions in the point obtain
r d /2 ^
V 0 y
we
I (n1 + n2 )ln^ + C.
\ r
= F
arcsin -
V1 - b7 k
\
, Ik,
+ 4* (k )N2, (23)
where N2 is an arbitrary integer. The system (20), (22), (23) determines the sought solution for the boundary conditions that we consider.
Boundary conditions 3.
r d /2 ^
On the surface of the antidot 2 6L = n/2 for a = 0, 6L = n/2 on the interval ,
V 0 y
(pL = (0 on the same interval in the point z = 0.
In an analogous way, from the boundary conditions on the surface of the antidot 2 we obtain
cVCl r n ln d + R + n ln R + C = F
V l0 y
V l0 y
. jr-b ,
arcsin-, k
+ 4* (k1 )N1, (24)
45
where N1 is an arbitrary integer. Similarly to the previous case, for the function P the relation P = {nl + n2 )ln--H C2 fulfils; from this relation,
2ln
I («1 + n2 )ln^ + C.
= F
arcsin -
Vl - b ' k
\
, k
+ 4K (k1 )N2 (25)
implies, where N2 is an arbitrary integer. From the boundary condition for yL we obtain
C3 = ®0 - П2Ж .
(26)
The system (24), (25), (26) determines the constants Ci, C2, C3, and together with (20) determines the sought solution.
5. COUPLE OF ANTIDOTS IN AN "EASY AXIS" ANTIFERROMAGNET
Let us select the functions P and Q in the following form:
P = — + n ln — + n2 ln
l l l0 l0
— - —0
w
+ C
V "0 Q = n1® + n2a + C3
(27)
where a is an azimuthal angle relative to the point r0. Such distribution corresponds, in particular, to a couple of circular antidots in an uniaxial antiferromagnet with the antiferromagnet of the "easy axis" type anisotropy. Similarly to the previous case we assume that antidots, in general, are not remote and that their radii R are the same, and we
( d \
choose the coordinate system in an analogous way, so that r0 =
V 0 y
functions P and Q on the antidots surfaces can be written as follows:
, d > 0. Therefore, the
D — 1 R
P = —h « ln —h n2 ln
l l l0 l0
d L R2 „R
—, 1 h—--2 — cos® ln\d2 d
Л
+ C
f
Q = «1® + «
ж - arctg
sin®
Л
(28)
d / R - cos®y
hC
on the surface of the antidot 1, and
f
£
P = — + n1 ln
l
d L R2 „R
—, 1 h—- + 2 — cosa L\ d2 d
Л
+ n2 ln
R
У
V l0 У
+ C
(29)
Q = «arctg I
sin a
d / R + cos a
+ « a + C
<
<
<
46
on the surface of the antidot 2. Analogously to the previous case we cannot set the
f r\
boundary conditions on the interval
y
but this time because of the dependence of the
function 0L on z, so we have to set at boundary conditions for the function 0L in some point. In a similar way to the previous case, it is convenient to choose the origin of the axis Oz in this point. Note that because of the presence of two constants C1 and C2 in the general form of the function 6L, in order to determine them we have to specify the function 6L in two points, and for the sake of convenience we can choose two points with the same z-coordinate and set the origin of the Oz axis in these points. Having made these remarks, we can now find the distribution of the antiferromagnetic vector in an analogous to the previous two cases way. Let us consider three analogous boundary conditions sets.
Boundary conditions 1.
On the surface of the antidot 2 6L = n/2 for a = 0, z = 0, yL = y0 for a = 0; on the surface of the antidot 1 6L = n/2 for y = 0, z = 0.
Solution (2), (27) in this case can be rewritten as
tg
(6
b
f
dn
"Vl^il
f
l
+ nx ln
r - exd
L
+ n2 ln
V ^ y
+ C,
ki
y y
(30)
Q = n^ + n2a + (p0
After substituting the boundary conditions into it we can see that the distribution constants C1, C2 are determined from the same system (21) as in the previous case.
Boundary conditions 2.
On the surface of the antidot 2 6L = n/2 for a = 0, z = 0, yL = y0 for a = 0; 6L = n/2 in
r d /2 ^
the point
0 0
v ~ J
From the boundary conditions for yL in a similar way to the previous boundary conditions choice we obtain C3 = y0, so that the solution of (2), (27) takes the form (30). The system for determining the distribution constants C1, C2 has the same form as for the boundary conditions 2 in the previous case, so these constants are determined from the system (22), (23).
Boundary conditions 3.
On the surface of the antidot 2 6L = n/2 for a = 0, z = 0; yL = y0 on the interval
r d /2 ^
and 0L = n/2 on the same interval in the point z = 0.
I 0 J
After substituting these boundary conditions into the solution (2), (27), we can see that despite of the different type of the antiferromagnet and the different form (27) of the distribution functions, constants of this distribution are determined by the similar to the previous case system (24), (25), (26).
47
z
6. PERIODIC ANTIDOT SYSTEM IN AN ISOTROPIC ANTIFERROMAGNET
Let us select the functions P and Q in the following form:
P = Z n in
f
r — r
+ C
Q = Tua'ni + C3
(31)
Such distribution corresponds, in particular, to a system of antidots with the centers {r0i} in a film composed of an isotropic antiferromagnet.
We assume that the radii of the antidots are all equal and equal to R. Let us consider the case when antidots in the antiferromagnet form a periodic structure with the period a along the Ox axis and the period b along the Oy axis. Therefore, the relation
ro, = ro + ae xP, + beyq,
(32)
is satisfied, here pi, qi are integers. Because of the translational symmetry of the system all the factors ni in the distribution (31) are equal: ni = n for any i.
Note that because of the periodicity of the elliptic functions we cannot consider the infinite sum of logarithms in the expression for the function P. However, we can consider the number of the antidots limited, but still large enough so that we can use the translational symmetry condition.
Let us use constant boundary conditions on the surface of some antidot: 6L and yL are
constant when
r - r
= R for the antidot i0. Note that in order to use such boundary
condition we should consider a periodic system of antidots that are distant from each other. Indeed, after substituting such boundary condition into the solution (31) we obtain
f
L
that depends on the point
the equation that contains a variable addend ^ ni ln
i*i0 V '0 J
of the antidot surface. However, if the antidots are distant from one another, so that the
r - r,
h
0,
condition
r — r
' 0i ' 0i0
>> R is fulfilled, this addend can be considered approximately
constant and we can write down the following relation on the surface of the antidot i0:
P » n ln
— + ln
l0 i^i,
f
r0i r0i0
R ~
+ C2 = n ln —h C2,
(33)
here C2 = ln
r,. — r
L
+ C2 « const. Note that these considerations are correct
for any antidot system with the antiferromagnetic vector distribution determined by (31), not necessarily a periodic system.
48
l
<
0
0
l
0
y
y
In a similar way, variable addend ^ ai (r )n enters boundary conditions for
9l.
This addend can be neglected provided ^Aa(i, i0 )<< —, where Aa(i, i0 ) is the
M0 2
maximum difference between the angles ai of the antidot i0 axis and an arbitrary point on the antidot i0 surface. In this case, due to the symmetry of the problem the sum
Y ai (r )n is a multiple of 2n for any point on the surface of the antidot i0, therefore, for
such antidots remoteness condition the following relation fulfils:
Q(r " »J = nai0 + C3. (34)
Thus, we can impose arbitrary constant boundary conditions on the surface of each antidot as long as these conditions are the same for all antidots. For example, we can choose the boundary conditions as follows:
Ol. (r - r„J = «K
Vl (r - roJ = R )=a,-0 + -±x
n 2
(35)
These boundary conditions correspond to a positive vortex distribution of the antiferromagnetic vector in the xy plane on the surface of each antidot. In this case
7T
C3 = — ± —, and the constant C1 can be determined from the condition
2
I
R
= F
o y
arcsin -
. Vi - b7
k
-, ki
+
4^(ki )N.
(36)
Here we put C2 = 0 (for a complete determination of the distribution we need either to
impose another boundary condition in addition to (35) or set one of the constants C1, C 2).
Note that the expression for 6L can be simplified considering the fact that if the case of the antidots remoteness is fulfilled, it can approximately be reduced to the sum over the four nearest antidots:
P = ln
f\r - r W
Y ' oi
+ C2 « n Z ln
x-y-y, |ib
f\r - r w
' ' Oi
+ C 2 -
-n
(
ln
V1 o y
r h
+ ln
V10 y
+ ln
( l~2— Va + b
= n
Z ln
x-X Sa.|y-y,\
r - rn.
yy
(„\ fu\
- ln
V1 o y
- ln
V 1o y
f l~2—Z2V\ Va + b
- ln
+ C
yy
(37)
49
o
o
1
1
o
o
1
o
1
1
o
o
In particular, for C2 = 0 the required distribution can be written
b • ctg | ~ \ = dn
(
cn^f1\ z
ln
r - r
(
- ln
V ^ J
- ln
V ^ J
(
- ln
2.1.2?! Ï
kl
Va2 + b
I
J J J
(38)
Vl = nZai + 7 i 2
with the constant C determined from the condition (36).
Note that the distribution we obtained is correct only far from the boundaries of the antidot system, so the approximation of the translational symmetry can be applied.
CONCLUSIONS
I
pa,|y-<b
0
Thus, we wrote down a solution of the Landau-Lifshitz equation for an antidot system in a film composed of an uniaxial or anisotropic two-sublattice antiferromagnet. Using this solution, we have found an antiferromagnetic vector configuration for a periodic remote antidot system in an isotropic antiferromagnet and for an antidot couple (in general, not a remote couple) in an isotropic antiferromagnet, in an antiferromagnet with uniaxial anisotropy of the "easy plane" type and in an antiferromagnet with uniaxial anisotropy of the "easy axis" type for three different variants of boundary conditions. We have shown that for an antidot system in a film composed of an uniaxial or isotropic two-sublattice antiferromagnet, constant boundary conditions on a surface of some (arbitrary) antidot is, in general, impossible, however, such boundary conditions are possible, for example, for a remote antidot system in an isotropic antiferromagnet.
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Горобець Ю. 1 Розподш вектора антиферомагнетизму для першдично!' системи вщдалених кругових антиточок у iзотропному антиферомагнетику та пари кругових антиточок у iзотропному або одновкному антиферомагнетику / Ю. 1 Горобець, О. Ю. Горобець, В. В. Кулш //
Вчет записки Тавршського нацiонального унiверситету iMeHi В. И. Вернадського. Серiя : Фiзико-математичн науки. - 2013. - Т. 26 (65), № 2. - С. 38-52.
Робота присвячена теоретичному дослщженню розподшу вектора антиферомагнетизму у плшщ з i30TponHoro та одновгсного aнтиферомaгнетикiв при нaявностi системи кругових антиточок. Отримано розв'язок рiвняння Ландау-Шфшица для вектора антиферомагнетизму у такш системi антиточок. Для перюдично1 системи вiддaлених антиточок у iзотропному антиферомагнетику, а також для пари антиточок (взагал^ не вiддaлених) у iзотропному, легковiсному та легкоплощинному антиферомагнетику отримано розподш вектора антиферомагнетизму за рiзних граничних умов. KnwHoei слова антиферомагнетик, мaгнiтнa тонка плiвкa, магштна антиточка, вектор антиферомагнетизму.
Горобец Ю. И. Распределение вектора антиферромагнетизма для периодической системы удаленных круговых антиточок в изотропном антиферромагнетике и пары круговых антиточок в изотропной или одноосном антиферромагнетике / Ю. И. Горобец, О. Ю. Горобец, В. В. Кулиш //
Ученые записки Таврического национального университета имени В. И. Вернадского. Серия : Физико-математические науки. - 2013. - Т. 26 (65), № 2. - С. 38-52.
Работа посвящена теоретическому исследованию распределения вектора антиферромагнетизма в пленке из изотропного и одноосного антиферромагнетиков при наличии системы круговых антиточек. Получено решение уравнения Ландау-Лифшица для вектора антиферромагнетизма в такой системе антиточек. Для периодической системы удаленных антиточек в изотропном антиферромагнетике, а также для пары антиточек (вообще, не удаленных) в изотропном, легкоосном и легкоплоскостном антиферромагнетике получено распределение вектора антиферромагнетизма при различных граничных условиях.
Ключевые слова: антиферромагнетик, магнитная тонкая пленка, магнитная антиточка, вектор антиферромагнетизма.
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Received 11 October 2013.
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