Adjustment of superconductivity and ferromagnetism in the few-layered ferromagnet–superconductor nanostructures
The phase diagrams of the few-layered nanosystems consisting of dirty superconducting (S) and ferromagnetic (F) metals are investigated within the framework of the modern theory of the proximity effect taking into account the boundary conditions. The F/S tetralayer and pentalayer are shown to hav...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1203462017-06-12T03:05:42Z Adjustment of superconductivity and ferromagnetism in the few-layered ferromagnet–superconductor nanostructures Izyumov, Y.A. Khusainov, M.G. Proshin, Y.N. К 100-летию со дня рождения Б.Г. Лазарева The phase diagrams of the few-layered nanosystems consisting of dirty superconducting (S) and ferromagnetic (F) metals are investigated within the framework of the modern theory of the proximity effect taking into account the boundary conditions. The F/S tetralayer and pentalayer are shown to have considerably richer physics than the F/S bi- and trilayer (due to the interplay between the 0 and π phase superconductivity and the 0 and π phase magnetism and nonequivalence of layers) and even the F/S superlattices. It is proven that these systems can have different critical temperatures and fields for different S layers. This predicted decoupled superconductivity is found to manifest itself in its most striking way for F/S tetralayer. It is shown that F/S/F/S tetralayer is the most perspective candidate for use in superconducting spin nanoelectronics. 2006 Article Adjustment of superconductivity and ferromagnetism in the few-layered ferromagnet–superconductor nanostructures / Y.A. Izyumov, M.G. Khusainov, Y.N. Proshin // Физика низких температур. — 2006. — Т. 32, № 8-9. — С. 1065–1077. — Бібліогр.: 36 назв. — англ. 0132-6414 PACS: 74.78.Fk, 85.25.–j, 74.62.–c, 85.75.–d http://dspace.nbuv.gov.ua/handle/123456789/120346 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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К 100-летию со дня рождения Б.Г. Лазарева К 100-летию со дня рождения Б.Г. Лазарева |
| spellingShingle |
К 100-летию со дня рождения Б.Г. Лазарева К 100-летию со дня рождения Б.Г. Лазарева Izyumov, Y.A. Khusainov, M.G. Proshin, Y.N. Adjustment of superconductivity and ferromagnetism in the few-layered ferromagnet–superconductor nanostructures Физика низких температур |
| description |
The phase diagrams of the few-layered nanosystems consisting of dirty superconducting (S) and
ferromagnetic (F) metals are investigated within the framework of the modern theory of the proximity
effect taking into account the boundary conditions. The F/S tetralayer and pentalayer are
shown to have considerably richer physics than the F/S bi- and trilayer (due to the interplay between
the 0 and π phase superconductivity and the 0 and π phase magnetism and nonequivalence of
layers) and even the F/S superlattices. It is proven that these systems can have different critical
temperatures and fields for different S layers. This predicted decoupled superconductivity is found
to manifest itself in its most striking way for F/S tetralayer. It is shown that F/S/F/S
tetralayer is the most perspective candidate for use in superconducting spin nanoelectronics. |
| format |
Article |
| author |
Izyumov, Y.A. Khusainov, M.G. Proshin, Y.N. |
| author_facet |
Izyumov, Y.A. Khusainov, M.G. Proshin, Y.N. |
| author_sort |
Izyumov, Y.A. |
| title |
Adjustment of superconductivity and ferromagnetism in the few-layered ferromagnet–superconductor nanostructures |
| title_short |
Adjustment of superconductivity and ferromagnetism in the few-layered ferromagnet–superconductor nanostructures |
| title_full |
Adjustment of superconductivity and ferromagnetism in the few-layered ferromagnet–superconductor nanostructures |
| title_fullStr |
Adjustment of superconductivity and ferromagnetism in the few-layered ferromagnet–superconductor nanostructures |
| title_full_unstemmed |
Adjustment of superconductivity and ferromagnetism in the few-layered ferromagnet–superconductor nanostructures |
| title_sort |
adjustment of superconductivity and ferromagnetism in the few-layered ferromagnet–superconductor nanostructures |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| topic_facet |
К 100-летию со дня рождения Б.Г. Лазарева |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120346 |
| citation_txt |
Adjustment of superconductivity and ferromagnetism in
the few-layered ferromagnet–superconductor
nanostructures / Y.A. Izyumov, M.G. Khusainov, Y.N. Proshin // Физика низких температур. — 2006. — Т. 32, № 8-9. — С. 1065–1077. — Бібліогр.: 36 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT izyumovya adjustmentofsuperconductivityandferromagnetisminthefewlayeredferromagnetsuperconductornanostructures AT khusainovmg adjustmentofsuperconductivityandferromagnetisminthefewlayeredferromagnetsuperconductornanostructures AT proshinyn adjustmentofsuperconductivityandferromagnetisminthefewlayeredferromagnetsuperconductornanostructures |
| first_indexed |
2025-07-08T17:42:18Z |
| last_indexed |
2025-07-08T17:42:18Z |
| _version_ |
1837101524275167232 |
| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9, p. 1065–1077
Adjustment of superconductivity and ferromagnetism in
the few-layered ferromagnet–superconductor
nanostructures
Y.A. Izyumov1, M.G. Khusainov2,3,4, Y.N. Proshin2,3
1 Institute of Metal Physics, Ural Division of RAS, Ekaterinburg 620219, Russia
2 Kazan State University, Kazan 420008, Russia
Email: Yurii.Proshin@ksu.ru
3 Max-Planck-Institute for the Physics of Complex Systems, Dresden 01871, Germany
4 «Vostok» branch, Kazan State Tupolev Technical University, Chistopol’ 422981, Russia
Received March 15, 2006, revised May 4, 2006
The phase diagrams of the few-layered nanosystems consisting of dirty superconducting (S) and
ferromagnetic (F) metals are investigated within the framework of the modern theory of the prox-
imity effect taking into account the boundary conditions. The F/S tetralayer and pentalayer are
shown to have considerably richer physics than the F/S bi- and trilayer (due to the interplay be-
tween the 0 and � phase superconductivity and the 0 and � phase magnetism and nonequivalence of
layers) and even the F/S superlattices. It is proven that these systems can have different critical
temperatures and fields for different S layers. This predicted decoupled superconductivity is found
to manifest itself in its most striking way for F/S tetralayer. It is shown that F/S/F�/S�
tetralayer is the most perspective candidate for use in superconducting spin nanoelectronics.
PACS: 74.78.Fk, 85.25.–j, 74.62.–c, 85.75.–d
Keywords: proximity effect, superconductivity, ferromagnetism, multilayers, critical temperature.
Introduction
The fabricated F/S heterostructures consisting of
alternating ferromagnetic metal (F) and supercon-
ducting (S) layers are well explored systems (see the
recent reviews [1–4] and references therein). Due to
the proximity effect, the superconducting order pa-
rameter can be induced in the F layer; on the other
hand, the neighboring pair of the F layers can interact
with each other via the S layer. Such systems exhibit
rich physics, which can be controlled by varying the
thicknesses of the F and S layers, or by placing the
F/S structure in an external magnetic field. Nu-
merous experiments on the F/S structures (contacts,
trilayers, and superlattices) have revealed nontri-
vial dependences of the superconducting transition
temperature Tc on the thickness of the ferromagnetic
layer df .
Buzdin et al. [5] were first who formulated the
boundary value problem for the pair amplitude (the
Cooper pair wave function) in a dirty superconductor
for the F/S superlattice. The critical temperature Tc
that was also calculated as the df function in Refs. 5, 6
exhibited both monotonic and nonmonotonic depen-
dences. Oscillations of T dc f( ) were related to periodi-
cal switching of the ground superconducting state be-
tween the 0 and � phases, so that the system chooses
the state with higher transition temperature Tc. In the
� phase state the superconducting order parameter �
in the neighboring S layers of the F/S superlattice
has the opposite sign, contrary to the 0 phase state in
which � has same sign for all S layers. The experimen-
tal evidence of the � superconducting state in the F/S
systems has been discussed in the review [2]. The con-
cept of a � junction was first proposed by Bulaevsky et
al. [7].
© Y.A. Izyumov, M.G. Khusainov, and Y.N. Proshin, 2006
In subsequent studies [8–11] the boundary condi-
tions have been derived from the microscopic theory,
and they are valid for arbitrary transparency of the
F/S interface. The solution of the boundary value
problem [8–15] has revealed an additional mechanism
of nonmonotonic dependence of Tc due to modulation
of the pair amplitude flux from the S layer to the F
layer. This modulation is caused by the change of the
FM layer thickness df . Moreover, it has also resulted
in a prediction of different types of behavior T dc f( )
such as reentrant [8–10,16] and periodically reentrant
superconductivity [8–10]. Note that both the oscilla-
tions and the reentrant behavior of T dc f( ) can appear
not only in the F/S superlattice but also in simple
F/S bilayer and F/S/F trilayer systems in which the
� phase superconductivity is impossible in principle!
The reentrant character of superconductivity that we
have predicted has been recently observed experimen-
tally in the Fe/V/Fe trilayer [17].
Now it may be considered as proven [1] that super-
conductivity in the layered F/S systems is a combina-
tion of the BCS pairing with a zero total momentum of
the pairs in the S layers and the pairing due to the
Larkin—Ovchinnikov—Fulde—Ferrell (LOFF) me-
chanism [18,19] with a nonzero three-dimensional
(3D) momentum of the pairs k in the F layer. The
LOFF pairs momentum k I/vf� 2 is determined by
the Fermi surface splitting caused by the internal ex-
change field I (where vf is the Fermi velocity in the F
layers). Usually it is assumed [5,6,8–15,20,21] that
the momentum of the LOFF pairs is directed across
the F/S interface (the so-called one-dimensional
(1D) case).
Of special interest is the study of the multilayered
F/S structures, in which various types of magnetic
order can arise in the F layers due to their indirect in-
teraction via the S layers. Recently the theory of the
proximity effect has been developed for the F/S
structures taking into account the inverse influence of
superconductivity on magnetism of the F layers and
on mutual orientation of their magnetizations. This as-
pect of the proximity effect has been studied for the
F/S/F trilayer «spin-switch» [22,23], and in the
F/S bilayer at exploring the possibility of the
cryptoferromagnetic state [24,25] and possible mag-
netic correlations acquired by a superconductor at the
S/F interface [26]. The long-range proximity effect
due to triplet superconductivity that arises in the case
of non-collinear alignment of magnetizations in the F
layers has been studied for the F/S/F trilayer system
in Refs. 27–30.
The multilayered F/S systems have additional
competition between the 0 and � phase types of super-
conductivity. Along to an interplay between the 0 and
� phase magnetism this leads to a new classification of
the F/S superlattice states [1,31,32]. In this classifi-
cation scheme the four different states (��) are distin-
guished by the phases of the superconducting (the
first symbol, �) and magnetic (the second one, �) or-
der parameters. So, the F/S superlattice possesses
two ferromagnetic superconducting (FMS) states (00,
�0), and two antiferromagnetic superconducting
(AFMS) ones (0� and ��). In the AFMS states the
phases � in the neighboring F layers are shifted by �,
i.e. the exchange fields I have opposite signs in the
neighboring F layers. This state with antiparallel
alignment of the corresponding magnetizations may be
considered as a manifestation of the � phase magne-
tism. Similar to the F/S/F trilayer [22,23], in the
case of the F/S superlattice the AFM ordering of the
magnetizations of all F layers leads to the significant
reduce of the pair-breaking effect of the exchange
field I for the S layers, and to the raise of the critical
temperature of the layered system. This theoretical
prediction has been experimentally confirmed for the
Gd/La superlattices [33]. Goff et al. have observed
that the superlattices with prepared antiferromagnetic
ordering of the magnetizations in the adjacent Gd lay-
ers undergo the transition into a superconducting state
at considerably higher temperatures in comparison
with the samples with ferromagnetic ordering magne-
tization of the Gd layers. This mutual accommodation
between the superconducting and magnetic order pa-
rameters reflects a quantum coupling between the
boundaries.
The F/S nanostructures possess two data-record-
ing channels: the superconducting one determined by
conducting properties of the S layers, and the mag-
netic one determined by ordering of the F layer
magnetizations. The F/S/F trilayer devices, pro-
posed in Refs. 22–24, operate through transitions be-
tween two states (the superconducting (S) and normal
(N) ones) that are induced by changes of the mutual
ordering of the magnetizations of the adjacent F lay-
ers. These changes are controlled by an external mag-
netic field H. The data stored in the superconducting
and magnetic channels of this switch device change si-
multaneously, and the magnetic order completely
determines the «superconducting information». The
scheme of a complex «spin device» with five possible
states on the basis of the F/S superlattices, in which
the superconducting and magnetic data-recording
channels can be controlled separately, has been pro-
posed in Refs. 31, 32.
However, both from the point of view of manufac-
turing and the «layer-by-layer» control by a weak
magnetic field, the few-layered systems with a limited
number of layers are more interesting objects. So, in
1066 Fizika Nizkikh Temperatur, 2006, v. 32, No. 8/9
Y.A. Izyumov, M.G. Khusainov, and Y.N. Proshin
Sec. 2 we theoretically investigate both well explored
F/S systems (bilayer, trilayers, and superlattice) and
scantily-known F/S tetra- and pentalayer (see
Fig. 1). We solve the Usadel equations for these struc-
tures taking into account the boundary conditions and
possible competition between the 0 and � phase mag-
netism and the 0 and � phase superconductivity. In
Sec. 3 we construct the phase diagrams with an opti-
mal set of parameters and discuss possible difference
of critical temperatures of different S layers.
2. Few-layered F/S structures
Some layered F/S structures have been studied ex-
tensively in recent years both experimentally and the-
oretically. Mainly the F/S bilayers (Fig. 1,a), the
F/S/F (Fig. 1,b), S/F/S (Fig. 1,c) trilayers, and
F/S superlattices were explored. The corresponding
references could find in the recent reviews [1,3,4]. In
this paper we consider both these systems and poorly
explored few-layered F/S systems: the F/S/F�/S�
tetralayer (Fig. 1,d) and the F/S/F�/S�/F� � pentalayer
(Fig. 1,e) with specific boundary conditions.
The proposed geometry of few-layered systems (see
Fig. 1) would allow us to simplify the solutions, and,
thereafter, to compare the obtained results for the
tetralayer and pentalayer structures with the ones for
the bilayer, trilayers and superlattice cases. The
primes indicate possible difference in properties of lay-
ers made of identical material.
To find the critical temperature we assume the
usual relation between the energsy parameters of the
system �F csI T�� ��2 , where �F is the Fermi energy
and Tcs is the critical temperature of the S material.
We also suppose the dirty limit conditions
ls s s�� �� 0, l af f f�� �� . Here l vs f s f s f, , ,
� is
the mean free path length for the S (F) layer; vs f, is
the Fermi velocity; �s f s f cs
/D / T, ,( )
2 1 2 is the
superconducting coherence length; a v / If f
2 is the
spin stiffness length; s0 is the BCS coherence length;
D v l /s f s f s f, , ,
3 is the diffusion coefficient. For sim-
plicity we will use the 1D model when both order pa-
rameters and the pair amplitude depend only on z
[1,21].
In this case, the common boundary value problem
[10] for each layer is reduced to the Gor’kov self-con-
sistency equations for the Gor’kov function F z( , )�
(note, in this paper we often use the term «pair ampli-
tude» for F z( , )� due to traditions [1] and for simpli-
city)
�
�
s s s
f f f
z T F z
z T F z
( ) Re ( , ),
( ) Re ( ,
�
�
�
�
�
�
2
2
0
0
� �
� �
�
�
).
(1)
and to the Usadel equations, that appear for the S
and F layers as follows
� �
�
— ( , ) ( ),
—
D
z
F z z
iI
D
z
s
s s
f
2
2
2
2
2
2
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
F z zf f( , ) ( ) .� �
(2)
In Eqs. (1), (2) � �
�T n( )2 1 is the Matsubara fre-
quency; � s f( ) and
s f( ) are the superconducting or-
der parameter and the electron-electron coupling con-
stant in the S(F) layers, respectively. The prime on
the summation sign indicates cutoff at the Debye fre-
quency �D. The diffusion coefficient Df in the F
layer is assumed to be real rather than complex [1]
since the difference between its two values is insignif-
icant under the conditions 2 1I f� �� used below (see
discussion in Ref. 21).
The coupling between the superconducting and fer-
romagnetic layers is provided by corresponding boun-
dary conditions, which connect the pair amplitude
fluxes with the pair amplitude jumps on the interfaces
of the layers, and are written in the following form
[8–10]
Adjustment of superconductivity and ferromagnetism in the few-layered nanostructures
Fizika Nizkikh Temperatur, 2006, v. 32, No. 8/9 1067
a b c
d e
Fig. 1. The geometry of various few-layered systems.
Bi-layer F/S (a), two trilayers F/S/F� (b) and S/F/S�
(c), tetralayer F/S/F�/S� (d), and pentalayer
F/S/F�/S�/F� � (e) are shown. Vertical arrows show the
directions (in-line) of the magnetizations that play the
role of the magnetic order parameter. Here z1 0� , and
thicknesses of outer F layers and S layers are equal to
d /f 2 and d /s 2, where df and ds are thicknesses of inner F
and S layers, correspondingly.
4 4
�
�
�
�
s s
s
s
z z f f
f
f
z z
v
D
F z
z v
D
F z
z
i i
�
�
�
�
�
�
�
�
�
�
� �
( , ) ( , )
� � �[ ( , ) ( , )].F z F zs i f i0 0� �� (3a)
Here index i numbers the inner interfaces. The upper
signs are chosen at the F/S boundaries, the lower
ones are chosen at the S/F boundaries. The pair am-
plitude fluxes through the outside boundaries
(z zleft
0 and zright , see Fig. 1) are absent
�
�
�
�
�
F z
z
F z
zz z z
( , ) ( , )
.
� �
left right
0 (3b)
In Eq. (3) � s f( ) is the boundary transparency at the
S(F) side (0 � � �� s f, ) [1,10]. They satisfy the de-
tailed balance condition: � �s s s f f fv N v N
, where
Ns f( ) is the density of states at the Fermi level.
The last conditions (3b) distinguish the case of
few-layered structures (Fig. 1) from the F/S super-
lattice case [1,35] in which the pair amplitude
F should satisfy to periodical conditions
F z L I F z Ii i( , ) ( , )�
e e� � , where L d ds f
� is the
superlattice period, and � and � are the phases of
superconducting and magnetic order parameters, re-
spectively.
In order to calculate the critical temperatures of
this F/S system taking into consideration the bound-
ary transparencies, thicknesses of layers, etc., we
should solve the system of equations (2) and (3) to-
gether with the self-consistency equations (1).
The powerful pair-breaking action of the exchange
field I (I Tcs�� � ) is the basic mechanism for the de-
struction of superconductivity in the F/S systems.
For simplicity [10] assume that
f
0 (� f
0) in the
F layers. We will search the solutions of equations
(1)–(3) for the inner layers as a linear combination of
symmetric and antisymmetric functions relative to the
centers of the corresponding S and F layers. These pair
amplitudes (Gor’kov functions) look the same as in
the superlattice case [32]. The zero flux of the pair
amplitude through the outside boundaries (3b) deter-
mines only the even cosine-like functions for the outer
layers. At these boundaries the antinodes should be
fixed.
Since we are mainly interested in performing quali-
tative studies of the properties of the few-layered
nanostructures, we will use the single-mode approxi-
mation to obtain the analytical solution of the compli-
cated boundary value problem. However, when quan-
titative estimates are needed (to fit theoretical results
to experimental data) the latter approximation works
well only for a certain range of the values of the pa-
rameters in the problem [13,14]. According to our es-
timates [10] the optimal set of parameters used below
is close to this range (ds f s f( ) ( )� ). Note, that in any
approximation (single-mode, multi-mode, etc. [14])
the symmetry of the problem determines the possible
solutions.
2.1. Simple few-layered systems (bi- and tri-layers)
and superlattice
Firstly let us consider the well-explored F/S
bilayer and two types of trilayers. The simplest solu-
tions of the boundary value problem (2), (3) for the
F/S bilayer (Fig. 1,a) can be written the following
form [1]
F B
k z d /
k d /
d / z
F A
k z
f
f f
f f
f
s
s
�
� � �
cos ( )
cos ( )
, ,
cos
(
2
2
2 0
— )
cos ( )
, .
d /
k d /
z d /s
s s
s
2
2
0 2� �
(4)
Here and below ks f( ) are the components of the wave
vector that describe spatial changes of the pair ampli-
tudes Fs f( ) across the layers (along the z axis) inde-
pendently of the frequency �. The chosen form of the
pair amplitudes is related to the symmetry of bilayer
and corresponding boundary conditions (3).
The complex values of wave vector kf for the ferro-
magnetic layer is defined as [1,21]
k
iI
Df
f
2 2
— . (5)
Substituting the solutions (4) into the self-consis-
tency equations (1) and performing the standard sum-
mation over � we derive the usual Abrikosov—Gor’-
kov type equation for the reduced superconducting
transition temperatures tc of the S layer
ln —Re ,t
D k
T tc
s s
cs c
�
�
�
�
! �
�
�
�
�
�
!
!
" "
1
2
1
2 4
2
�
(6)
where t T /Tc c cs
; "( )x is the digamma function,
and the pair-breaking parameter D ks s
2 is the solution
of the other transcendental equation, that is deter-
mined from the condition of nontrivial compatibility
of set of linear equations for factors B, A. In turn this
set can be obtained by substitution of the solutions
(4) in (3). In this way we obtain one possible state
for the F/S bilayer case
#$ �
1 0, (7)
where
#
�
$
�
�
� �
4
2
1
4
2
1
D k
v
k d D k
v
k ds s
s s
s s f f
f f
f f
tan tan, .
(8)
1068 Fizika Nizkikh Temperatur, 2006, v. 32, No. 8/9
Y.A. Izyumov, M.G. Khusainov, and Y.N. Proshin
Equation (7) is the sole equation on ks which does
not depend on the orientation of magnetization in the
F layer. This expression (7) can be rewritten in
long-known form [1]
D k
k d v
v
D k
k d
s s
s s s s
s f
f f
f f
00
00
2
4
2
tan
cot
�
�
�
�
�
�
!
!
�
�
. (9)
Found from Eq. (7) or (9) the ks wave vector being
substituted to (6) allow us to calculate the Tc of
bilayer. It leads to the well known 00 type of solution
(according classification [1] the 00 state is one with
the 0 phase superconductivity and the 0 phase magne-
tism).
For the F/S/F� trilayer we should change second
expression in Eq. (4) for the inner S layer, adding the
sin-like summand, and add new expression for the
outer F� layer. Thus, three functions are
F B
k z d /
k d /
d / z
F A
k z
f
f f
f f
f
s
s
�
� �
cos ( )
cos ( )
, — ,
cos (
2
2
2 0
�
�
�
� �
d /
k d /
C
k z d /
k d /
z d
s
s s
s s
s s
s
2
2
2
2
0
)
cos ( )
sin ( )
sin ( )
,
,
cos ( )
cos ( )
, .F B
k z d d /
k d /
d z d d /f
f s f
f f
s s f�
�
� � �
�
� � �
2
2
2
(10a)
(10b)
(10c)
Recall, that in the paper the quantities related to the
different S and F layers layer are denoted by the
primes in correspondence with Fig. 1. In Eq. (10b)
the first term is responsible for the symmetric 0 phase
solution, while the second term is responsible for the
appearance of antisymmetric � phase solution.
Without loss in generality we assume that the mag-
netization in the left outer F layer is always upward in
all considered F/S systems (Fig. 1). Then, in the
framework of the made approximations, the complex
values of wave vectors kf and kf� for the F and F� lay-
ers, correspondingly, with the upward magnetizations
are defined as [1,21]
k k
iI
Df f
f
2 2 2
�
�( ) . (11a)
Thus, the Eqs. (11a) are valid for the case of the mu-
tual ferromagnetic (FM) ordering of the
magnetizations in all ferromagnetic layers. When we
have antiferromagnetic (AFM) configuration with
magnetization M� oriented «downward» (I I�
� , i.e.
the phase � of the magnetic order parameter equals
�), the value of kf� for the F� layer can be found from
following expression
( ) ( ) ,*k
iI
D
kf
f
f
�
2 22
(11b)
where kf is defined by Eq. (11a).
From condition of nontrivial compatibility of set of
linear equations for factors B, A, C, B� we obtain two
different solutions. One of them is the 00 state for FM
ordering of magnetizations of the adjacent F and F�
layers, which coincides with above stated solution for
the F/S bilayer (7). Another one is the 0� state which
corresponds the 0 phase type of superconductivity and
� phase type of magnetism (i.e., AFM ordering of
magnetizations of the adjacent F and F� layers)
� � � �#% $ % # $2 1�
— Re , (12)
when wave vector kf� in the F� layer is complex conju-
gate to kf in the F layer (see Eq. (11b) above). In
Eq. (12) we use this fact, that leads to $ $�
*, and in-
troduce denotation
%
�
�
4
2
1
D k
v
k ds s
s s
s scot . (13)
The critical temperatures corresponding to both states
can be calculated with Eq. (6) after substitution of
ks
00 found from (8) and ks
0� found from (12).
To obtain the set of solutions for the S/F/S�
trilayers (Fig. 1,c) we have to simply exchange the in-
dices and factors in Eqs. (10): s f& ; and B A' ,
B A� ' �, A B' , and C D' .
The states of the S/F/S� trilayer are independent
of orientation of magnetization in the F layer, it corre-
sponds to the 0 phase of magnetic order parameter. In
this case we formally have two equations for critical
temperatures for layer S (see Eq. (6)) and for layer S�
(to get an equation for tc� it is necessary to exchange tc
for t T /Tc c cs�
� and ks for ks� in Eq. (6)). The critical
temperatures depend heavily on the relative sign of
the superconducting order parameter � in the S and S�
layers. So, the two known states are obtained. The
first one is the 00 sate defined by Eq. (8) which is
identical for both S and S� layers (hence # #�
, then
T Tc c
�
). The second solution is the �0 state
#( # (�
) � �
) �1 0 1 0; , ; ,layer S layer S (14)
where denotation ( is
(
�
�
4
2
1
D k
v
k df f
f f
f f
cot . (15)
The Eqs. (14) also lead to the # #�
, then k ks s�
,
and hence we also have common critical temperature
in the �0 state for whole sample, i.e. T Tc c�
. Note,
the wave vector ks in the �0 state can be found from
Eq. (9) by substituting ks
�0 for ks
00 in the left-hand
Adjustment of superconductivity and ferromagnetism in the few-layered nanostructures
Fizika Nizkikh Temperatur, 2006, v. 32, No. 8/9 1069
side and � tan ( )k d /f f 2 for cot ( )k d /f f 2 in its
right-hand side.
Before a consideration of the F/S tetralayer and
pentalayer we shortly discuss the F/S superlattice
case, in which the full competition between the 0 and
� phase types of superconductivity and magnetism is
possible [1,31,32,35]. The pair amplitudes for all S
layers will be analogous to one for inner S layer in the
F/S/F trilayer (10b), the ones for all F layers will
correspond to one for inner F layer in the S/F/S
trilayer. Following to the cited papers the supercon-
ducting states of the F/S superlattice can be classi-
fied using four different sets ��: 00, 0�, �0, and ��.
Corresponding wave vectors ks for three of these
states are already defined by Eqs. (7), (12), (14), cor-
respondingly. The last �� state is determined by ks ,
which is found by following equation [1,31,32]
� � � �#%( % # (2 1�
— Re . (16)
Note, that the competition between the 0 phase super-
conductivity generated by Eq. (7) and the � phase su-
perconductivity generated by Eq. (14) for the F/S
superlattice was firstly proposed fifteen years ago
[5,6] to explain nonmonotone experimental T dc f( )
dependence. A manifestation of � phase superconduc-
tivity in the experiments on the Josephson current
can be found in the review [2].
2.2. Tetra- and penta-layers
For the F/S/F�/S� tetralayer (see Fig. 1,d) we
have the following form of solutions
F B
k z d /
k d /
d / zf
f f
f f
f
�
� �
cos ( )
cos ( )
, — , ( )
2
2
2 0 17a
F A
k z d /
k d /
C
k z d /
k ds
s s
s s
s s
s
�
�
�cos ( )
cos ( )
sin ( )
sin (
2
2
2
s
s
/
z d
2
0 17
)
,
, ( )� � b
F B
k z d d /
k d /
D
k z d
f
f s f
f f
f s
�
�
� � �
�
�
� �
� �
cos ( )
cos ( )
sin (
2
2
�
�
� � �
d /
k d /
d z d d
f
f f
s f s
2
2
17
)
sin ( )
, ( ), ( )c
F A
k z d d /
k d /
d d z d
s
s f s
s s
f s f
�
�
� � �
�
� � �
cos ( )
cos ( )
,
( ) (
3 2
2
� 3 2 17d /s ). ( )d
At last for the F/S/F�/S�/F� � pentalayer (see
Fig. 1,e) in comparison with Eqs. (17) we should re-
place the equation (17d) on the equation (18a) and
add the equation (18b)
F A
k z d d /
k d /
C
k z d
s
s f s
s s
s
�
�
� � �
�
�
� �
� �
cos ( )
cos ( )
sin (
3 2
2
f s
s s
f s f s
f
f
d /
k d /
d d z d d
F B
k
�
�
� � � �
��
� �
��
3 2
2
2
)
sin ( )
,
cos ( )
cos ( )
,
.
z d / d
k d /
d d z d / d
f s
f f
f s f s
� �
��
� � � �
3 2 2
2
2 3 2 2
(18a)
(18b)
So the full set of solutions for the considered
pentalayer includes Eqs. (17a), (17b), (17c), (18a),
(18b). Recall that according to our denotations all
quantities related to the F� � layer are denoted by the
double primes. Once again, for convenience we as-
sume that the magnetization in the left F layer is al-
ways upward (Fig. 1).
Naturally, by symmetry, for the bilayer, both
trilayers and superlattice, studied before (see Refs. 1,
2, 4 and references therein) all superconducting layers
are placed in identical surroundings. The same we can
say about symmetrical configurations of the
F/S/F�/S�/F� � pentalayer. In the end this leads to
the identical superconducting properties of all the S
layers, that is, all S layers have common critical tem-
perature. In contrary, for the F/S/F�/S� tetralayer
and F/S/F�/S�/F� � pentalayer (asymmetrical case)
the solutions will be different for the different S and S�
(F and F�) layers. This fact reflects the general prop-
erty of these systems: the nonequivalence of layers of
the same type, which results in different supercon-
ducting properties of the S and S� layers.
For the F/S/F�/S� tetralayer the nonequivalence
of the S(F) and S�(F�) layers are essential. In common
case they have different local surroundings. The wave
vectors ks for the S layer and ks� for the S� layer are de-
termined from the condition of nontrivial compatibil-
ity of the corresponding set of equations on the factors
B, A,C, B�, D�, A�, which can be obtained after substi-
tution of the expressions for pair amplitudes (17) into
Eqs. (3). It is possible to factorize the corresponding
determinant and to obtain the following equation
( )[( )( — ) ( )( )]
( )[(
# ( #$ %$ %$ #$
# $ #
� � � � � � � � � �
� � � �
1 1 1 1 1
1 $ %( %$ #(� �� � � � �
1 1 1 1 0)( ) ( )( )] .
(19)
Equation (19) can be simplified by taking into ac-
count the independence of the solutions for the S and
S� layers and knowing the solutions for the
superlattice case. It is possible to obtain the following
1070 Fizika Nizkikh Temperatur, 2006, v. 32, No. 8/9
Y.A. Izyumov, M.G. Khusainov, and Y.N. Proshin
sets of equations on ks and ks� , which are different for
the FM and AFM configurations. Note that only
equations leading to the finite nonzero critical tem-
perature are kept in these sets (see Eqs. (20) and (21)
below).
For the FM ordering of the magnetizations we ob-
tain two cases FM (a a, �) and FM (b b, �)
FM
layer S
layer S
a
a�
�
�
��
�
�
�� �
� �
�
� � �
�
�
1 0 00
1 0
; ( )
; ( )
( )( ) ; (
00
2 2
�
�
�
�
�
�
�
��
�
�
�� �
�
� �FM layer Sb
b
�
�� � �
� �
� � �
0
1 0 0
~)
; ( ) .�
� � � �
�
�
�
�� layer S
(20)
Here $� and (� for the F� layer are substituted by $ and
(, respectively, due to Eq. (11a). We see that cases
FM(a) and FM(a�) coincide with known 00 solutions
for layers S and S�, correspondingly. The FM(b�) one
for the outer S� layer is the �0 solution also known for
the S/F/S trilayer (14) and superlattice case. The
FM(b) for the inner S layer is completely new solu-
tion. Its presence is related to the external boundary
conditions (3b) since the pair amplitudes (17a),
(17d) contain only even cosine solutions in contrary
to the F/S superlattice case. This state is the new �
superconducting state, and in order to distinguish it
from the previous superlattice solutions we will de-
note the new ones with tilde ��~ (� �
0, is the phase
of the magnetic order parameter). Thus the FM (b)
state can be denoted as the �0~ state of the tetralayer.
According to equation (11b), when k kf f
�
* (lead-
ing to $ $�
* and ( (�
*), for the AFM ordering of
magnetizations in the tetralayer we have two other �
magnetic cases AFM(c c, �) and AFM(d d, �)
� � � �
AFM
layer Sc
c�
�
�
��
�
�
� � �
� � �
��� � � � �
� �
2 1 0
1 0
Re ; ( )
* ; ( )
( )(*
�
�
�
�
�
��
�
�
�
��
�
�
� � �
layer S
AFM
00
2d
d
���� � � � � ��
� � �
*
*
) ; ( )
; ( ) .
~�
� � �
�
�
�
�
�
��
2
1 0 0
layer S
layer S
(21)
In this case we also have three solutions formally
coinciding with ones from known classification
scheme [1,31,32]: AFM(c�) = (00), AFM(c) =
= (0�), AFM(d�) = (�0). The AFM(d) solution is
the new ��~ one.
In the general case, there are 4 different solution
sets FM(a a, �), FM(b b, �), AFM(c c, �), and AFM(d d, �)
for the S and S� layers, each of which completely de-
fines the state of both layers and, hence, the corre-
sponding reduced transition temperatures tc and tc�
(6). However, since the solution tc of Eq. (6) does not
change when ks is replaced by its complex conjugate
the solutions tc� for the S� layer do not depend on rela-
tive orientation of the magnetizations: the solution of
the equation (6) for the FM(a�) case coincides with
the solution for the AFM (c�) case. The same is true for
the solutions for the FM(b�) and AFM(d�) cases. Thus,
we have only two distinguishable solutions a� and b�
for the S� layer.
The latter can be easily understood from the physi-
cal point of view. Only one ferromagnetic layer (F�)
acts on the outer S� layer. As a result the state of the
layer depends only on the magnitude of the exchange
field in the F� layer and does not depend neither on its
sign nor on mutual ordering of the magnetizations. In
other words, the S� layer is always in the local
ferromagnet (FM) environment, therefore the � mag-
netic solutions do not exist for this layer.
The F/S/F�/S�/F� � pentalayer (see Fig. 1,e) may
have three nonequivalent configurations in which the
S and S� layers are in essentially different local envi-
ronment. There are two symmetrical configurations:
the first one is completely ferromagnetic (FM), when
the magnetizations of all three F layers have the same
directions (we designate this case as *S*S� *, where ar-
rows show the direction of magnetizations M, M�, and
M� �); and the second case is antiferromagnetic (AFM
or *S+S� *, as shown in Fig. 1,e). The third case is the
nonsymmetrical one (*S*S� +), and for this mixed case
we introduce the FMAFM designation.
In the first symmetrical cases we have two sets
FM(a,a�) and FM(b,b�) of solutions (for the pen-
talayer states notation we use underlined letters).
By symmetry, into each set we have the solutions coin-
ciding for the S and S� layers. Actually, the FM(a) =
= FM(a�) solutions are equal to the 00 one, defined
by (7), and they coincide with FM(a) = FM(a�) =
= AFM( )c� solutions for the tetralayer (see Eqs. (20),
(21)). In turn, the FM(b) = FM(b�) solutions are
equal to the �0
~
one, and they coincide with FM(b) so-
lution for the inner S layer of tetralayer, defined in
(20).
In the second symmetrical *S+S�* case we also have
two sets AFM(c c, �) and AFM(d d, �) solutions coincid-
ing into each set for both S and S� layers. So, the
AFM(c) = AFM(c�) solutions are equal to (0�) one,
defined by (12), and they coincide with AFM(c) solu-
tion for the inner S layer of tetralayer (21). Two other
solutions AFM(d) = AFM(d�) = ( )~�� = AFM(d), the
last one is defined in Eq. (21) for the inner S layer of
tetralayer too.
At last, in the nonsymmetrical *S*S� + case we have
two sets FMAFM (e,e�) and FMAFM (f,f �) of non-
equivalent solutions which coincide with correspond-
ing ones for tetralayer (20), (21):
Adjustment of superconductivity and ferromagnetism in the few-layered nanostructures
Fizika Nizkikh Temperatur, 2006, v. 32, No. 8/9 1071
FMAFM layer S FM( FM( AFM(
FMAFM
( ) ( ) ) ) ),
(
e a a c
e
�
� � � �
�
00
) ( ) ( ) ( );
( ) ( )~
�
�
� � �
�
�
�
0
0
layer S FM AFM
FMAFM layer
b d
f S FM(
FMAFM layer S AFM
� �
�
b
f d
),
( ) ( ) ( ).~��
Thus, the expressions obtained above include a com-
petition between the 0 phase and the � phase types of
superconductivity. They also take into account inter-
action of the localized moments of adjacent ferromag-
netic layers through the superconducting layers.
In the next section we will examine the obtained
solutions and clarify the winners in the interplay of
the possible states.
3. Discussion of the phase diagrams of layered
F/S systems
The set of equations (6)–(21) allows us to investi-
gate the dependence of critical temperatures (tc or t c� )
of the layered F/S systems on the reduced thickness
of the magnetic layer d /a df f
~
.
Keeping in mind a possibility of an application of
the system as a «spin device» we have searched for
such a set of parameters for which the difference be-
tween the various states of studied systems is suffi-
ciently large to be observed. After performing numer-
ous computer experiments we have found a range for
the values of the parameters that satisfies these condi-
tions. The optimal range of parameters should be as
follows: the boundary should be sufficiently transpar-
ent (� s , ��5 1), the ferromagnetic metal should be
sufficiently dirty or (and) weak enough in regard to
its magnetic properties (2 015 1I l /af f f�
��� . ),
and the parameter n N v /N vsf s s f f
� 1. From exper-
imental viewpoint this constraint on the values of the
parameters does not look unreasonable.
A set of phase curves t dc f( ) for bilayer, trilayers,
and superlattices for the optimal values of the parame-
ters is shown in Fig. 2. These systems are well ex-
plored (see, for example, Ref. 1 and references
therein), but this picture allows us to explain the
phase diagram of all stated layered F/S systems with
unified positions and to compare them with each other
and with ones for the F/S tetra- and pentalayer (see
Figs. 3, 4 below).
So, the F/S bilayer (Fig. 2,a) has only possible so-
lution (00). Note, that critical temperature has
reentrant behavior. The same type of T dc f( ) depend-
ence was firstly predicted by us [8,9], and it was ob-
served in the Fe/V/Fe trilayer not long ago [17].
In the F/S/F trilayer (Fig. 2,b) the competition
of two magnetic states takes place. Due to partial
compensation of paramagnetic effect of exchange field
I the � magnetic state (0�) has higher critical temper-
ature than the 00 state. Therefore this 0� state wins
and, according to the theory of second order phase
transitions, the F/S/F trilayer always is in AFS con-
dition at T Tc� ( )0� . But if we turn on a weak exter-
nal magnetic field H greater than coercive field,
Hcoer , we can go the trilayer into the FM state in
which the magnetizations of both F layers are parallel,
and the system becomes normal. As noted earlier
[22,23], this system could be used for superconducting
«spin switch» construction with two possible states.
As to the S/F/S trilayer (Fig. 2,c), the spontane-
ous transitions take place with increasing thickness of
the F layer. The first one is from the 0 superconduct-
ing state (00), when both S layers have the same sign
of superconducting order parameter �, to the � super-
conducting state state (�0), when � in the S layers
differ by sign, at
~
.d � 0 4. The reverse transition is at
~
.d � 12. The such trilayers are often used for experi-
mental investigation of so called the � junctions by
1072 Fizika Nizkikh Temperatur, 2006, v. 32, No. 8/9
Y.A. Izyumov, M.G. Khusainov, and Y.N. Proshin
a b
c d
Fig. 2. The phase diagrams (t dc �
~) of the F/S
nanostructures for the following values of parameters:
� s � 15, 2 01I f� � . , nsf � 14. , ls s� 025 0. � , and ds s� 072 0. � .
In the figure t T Tc cs� / is the reduced temperature, and
~ /d d af f� is the reduced F layer thickness. The phase di-
agram of the F/S bilayer with one possible state (00)
(a); the phase diagram of the F/S/F trilayer for the FM
(00) and AFM ( )0� configurations (b); the phase diagram
of the S/F/S trilayer with competition between 0 phase
and � phase of superconductivity (c); the phase diagram of
the F/S superlattice where the thicknesses of all F layers
equal df , and the thicknesses of all S layers equal ds (d).
the Josephson current methods (see, for example,
Ref. 2).
At last we see all four states in the phase diagram
for the F/S superlattice (Fig. 2,d) for which, prop-
erly speaking, the classification scheme has been pro-
posed [1,31,32]. In cited references the detail discus-
sion of the properties of such multilayered system
connected with competition between the 0 and � phase
superconductivity and the 0 and � phase magnetism
can be found. Emphasize, that in the AFM condition
due to partial compensation of paramagnetic effect of
exchange field the F/S superlattice has the higher Tc
than one in the FM state. The complicated phase dia-
gram of the F/S superlattice allowed us to propose a
spin device with five possible states controlled by
small external magnetic field [35].
Completing the short review of phase diagram of
studied earlier F/S system, we stress that all layered
F/S structures presented in Fig. 2 posses single criti-
cal temperature Tc for all S layers.
3.1. Decoupled superconductivity in the F/S tetra-
and pentalayer
The set of phase diagram for the F/S/F�/S�
tetralayer is presented in Fig. 3.
As one might expect, the a� and b� curves for the S�
layer in Figs. 3,a and 3,b are identical for both FM
and AFM configurations, and the a� curve for the S�
layer coincides completely with the a curve for the S
layer since both of them describe the same 00 state.
The rest of the states for the inner S layer (the b curve
for the FM configuration, and the c and d curves for
the AFM one) have different dependencies as com-
pared with the ones for the S� layer and each other.
Hence as it follows from Figs. 2, 3 and discussion
presented here the tetralayer has more physically dif-
ferent states than the F/S/F trilayer and even the
F/S superlattice!
Thus, the 00, �0, and 0� states of the four-layered
system (Figs. 3,a,b) correspond to the same states of
the superlattice (Fig. 3,c). The �0~ state (the b curve in
Fig. 3,a) and the ��~ one (the d curve in Fig. 3,b) are
the states associated with the � phase superconductiv-
ity. These are the new solutions which are not found
in the superlattice case (Fig. 2,d). The main differ-
ence between the new ��~ and the known �� states is
the peak position. For the inner S layer it is shifted to
the lower values of the df thickness compared with
the superlattice case due to the implementation of the
external boundary conditions (3b).
The above-mentioned peculiarities of the four-lay-
ered system lead to different critical temperatures of
different S layers. To show this consider the FM con-
figuration (Fig. 3,a) in detail. If there is no difference
between tc and tc� for the 00 state then the case of the �
phase superconductivity is more interesting since there
is a difference between tc� ( )�0 and tc( )~�0 . Actually for
each superconducting layer the upper envelope curve
is realized due to free energy minimum condition. In
the case of the FM configuration that will be a b a�� �� �
curve and a b a� � one for the S� and S layers, respec-
tively. This leads to switching the ground state be-
tween the states with the 0 and � superconducting
phases as the thickness
~
d changes just as it was shown
for the S/F/S layer in Fig. 2,c (at
~
.d � 0 4 and
~
.d � 12, respectively).
In the � phase superconductivity case, the order pa-
rameter � has opposite signs for the S and S� layers.
Accordingly, the pair amplitude in the inner F� layer
(Eq. (17c)) has a sine-like behavior (B�
0) and is
Adjustment of superconductivity and ferromagnetism in the few-layered nanostructures
Fizika Nizkikh Temperatur, 2006, v. 32, No. 8/9 1073
a b
c d
Fig. 3. The phase diagrams (t dc �
~) of the F/S/F�/S�
tetralayer for the same set of parameters as in Fig. 2. The
tc� curves for the outer S� layer are denoted using letters
with a prime. The letters without a prime indicate the tc
curves for the inner S layer. All denotations of curves cor-
respond to ones entered in Eqs. (20) and (21). In the pan-
els a and b the arrows show the ( )t tc c� � difference be-
tween the states which are discussed in the paper. The
phase diagram for the FM configuration of the
magnetizations of both F layers (a); the phase diagram for
the AFM configuration (b); the combined phase diagram
of the F/S/F�/S� tetralayer (c); the generalized phase
diagram of the tetralayer (d). Vertical arrows show the
direction of the magnetization in the corresponding ferro-
magnetic layer. The letters S and N stand for the super-
conducting and normal states of the superconducting lay-
ers, respectively.
antisymmetric with respect to the layer center at
which the sign change of the pair amplitude takes
place while traversing the F� layer. The above-men-
tioned different Tc behavior in the S and S� layers (the
b and b� curves in Fig. 3,a, respectively) leads to a dif-
ference between critical temperatures tc and tc� . For
instance, at
~
.d
15 the reduced critical temperature of
the S layer tc is equal to 0.177, and tc�
0163. , at
~
.d
0 5 the difference is larger: tc�
0 308. and tc
016. .
If the reduced thickness
~
d were equal 0.6, the differ-
ence would be almost maximal: tc�
0 346. , and
tc
0154. .
The reduced critical temperatures tc and t c� that
correspond to these three values of the reduced thick-
ness
~
d are shown in Fig. 3,a by arrows. The difference
between two critical temperatures tc and tc� should be
observed in experiments with the special field-cooled
samples prepared with the FM ordering of the
magnetizations (see Ref. 33, for experimental details).
The appearance of the critical temperature differ-
ence for the F/S tetralayer is a manifestation of the
critical temperatures hierarchy in the clearest form.
The origin of the Tc difference is obvious because,
firstly, the S and S� layers are situated in different
magnetic environment and, secondly, they have differ-
ent boundary conditions. In particular it is expressed
in the above mentioned shift of peak of the new �0~ de-
pendence due to the outside boundary conditions.
Note, that for the same reason the critical fields Hc
and H c� will differ for layers S and S�, correspond-
ingly.
For the AFM configuration of the F/S/F�/S� sys-
tem we have a similar picture (Fig. 3,b), but in this
case there are four different curves. Note that all
above mentioned peculiarities take place as well. As it
has been discussed above, the phase curves for the S�
layer are the same for both the FM and the AFM ori-
entations. Two different solutions are obtained for the
inner S layer. One of them is the known «super-
lattice» solution 0� (curve c) while the second one is
the new ��~ solution (curve d). There is also a competi-
tion between the 0 and � phase superconductivity that
leads to a appearance of the corresponding envelope
curves of the second order phase transition for the S
and S� layers (c d c� � and a b a�� �� �, respectively).
The 0� solution corresponds to the D�
0 and B� - 0
case, and the A and C factors are not equal 0, i.e., the
pair amplitude in the S layer does not possess any par-
ity. The admixture of the sine solutions to the cosine
ones in expression (17b) reflects the partial compensa-
tion of the paramagnetic effect of exchange field I for
the S layer in the AFM state with antiparallel align-
ment of the F layers magnetizations. The previous
statement applies to the ��~ state in the S� layer too.
As in stated above FM case, the difference between
tc and tc� can be observed in experiments with the spe-
cial field-cooled AFM samples.
Let us take up the common case, when there is the
interplay of all the four states (20), (21), to clarify
the winners in this competition. For convenience all
the phase curves are shown in Figs. 3,a,b in one com-
bined diagram (Fig. 3,c).
Note, that at
~
.d
0 5 the ��~ state (the d curve in
Fig. 3,b and in Fig. 3,c) has the highest Tc among all
possible states for the S layer tc � 0 25. , but that is
lower than the appropriate temperature for the �0
state of the S� layer tc� � 0 31. . According to the theory
of second-order phase transitions, the state possessing
the lower free energy (higher Tc) is realized. Thus for
the samples with the reduced thickness
~
.d
0 5 the S
and S� layers are both in the normal (N) state if the re-
duced temperature t tc� � � 0 31. . Below tc� the S� layer
becomes superconducting (S) but the S layer remains
in the N state while t tc� � 0 25. . Finally, at t tc� the
AFMS state (AFM(d d b, � . �)) wins and for the whole
system we have the case with the � phase superconduc-
tivity and the � phase magnetism.
At
~
.d
15 we have the following chain of
the second order phase transitions: *N+N (or *N*N)
tc � 027./ '//// *S+N tc � 019./ '//// *S+S (see a caption
to Fig. 3,d for an explanation of the notation). Thus
at low temperatures the AFMS state (AFM(c c a, � . �))
wins too, but this state is associated with the 0 phase
superconductivity. Note in the framework of our the-
ory only transition temperatures can be found and it is
not possible to determine what state inside the «nor-
mal» state region is preferable.
The analogous analysis can be carried out for the
entire range of the reduced F layer thicknesses
(
~
)0 2� �d . Assume the system can choose its own
state according to the theory of the second order phase
transitions. The state with higher critical temperature
wins, and one of four states defined by Eqs. (20), (21)
(see also Figs. 3,a,b,c) is realized for the system. A
complete phase diagram constructed for the system is
presented in Fig. 3,d. Four different regions can be de-
fined for this diagram: at high temperatures both S
and S� layers are in normal state and the mutual order-
ing of the magnetizations in the F and F� layers is un-
important. As it follows from the phase diagram, there
are two regions marked in dark grey color with de-
coupled superconductivity} in antiferromagnetic state
for which the inner S layer is superconducting (S),
and the outer S� one is normal (N). The striped marked
region also corresponds to the state with decoupled
superconductivity when the outer S� layer is supercon-
ducting and the inner S layer is normal. Finally, at
low temperatures and/or at small df thicknesses the
1074 Fizika Nizkikh Temperatur, 2006, v. 32, No. 8/9
Y.A. Izyumov, M.G. Khusainov, and Y.N. Proshin
system is in the ground AFMS state (grey marked re-
gion).
Thus, we can predict, there are distinct regions of
parameters when the F/S/F�/S� tetralayer should be
in the state with decoupled superconductivity. More-
over, if the inner S layer is in the superconductive
state then ordering of the magnetizations should be
antiferromagnetic. This is the result of the inverse ac-
tion of superconductivity on magnetism.
At last let us consider the F/S pentalayer case
shown in Fig. 4. Note that the tetralayer case more
comprehensively since it has the extra �0 state in com-
parison with the pentalayer one. This leads to that the
F/S pentalayer possesses simpler phase diagrams.
Naturally, in the symmetrical cases the both S and S�
layers have identical critical temperatures T Tc c
� .
In the FM case (Fig. 4,a) the both S layers may be
simultaneously either in the 00 state (curve a a
�) or
in the �0 state (curve b b
�), that correspond to the
( , )a a� and b� curves for the tetralayer case in Fig. 4,a,
respectively. Competition between the 0 and � phase
superconductivity leads to switching ground state as it
was shown in the tetralayer case above.
In the AFM case (Fig. 4,b), by symmetry, we have
single Tc for all system too, and the states of the both
S layers are also determined simultaneously either the
0� one (curve c c
�) or the ��~ one (curve d d
�).
These states correspond to the c and d curves for the
inner S layer in Fig. 3,b for the tetralayer, accor-
dingly.
In the nonsymmetrical FMAFM case each of the
superconducting layers is situated in different local
magnetic surroundings, and we have a competition be-
tween the (e, e�) set and (f, f �) set (see Fig. 4,c). The 0
phase superconducting state (e, e�) wins at
~
.d � 0 4 and
~
.d � 12. The � phase superconducting state (f, f �) is
energetically preferable at 0 4 12.
~
.� �d conditions. In
any case the critical temperature T c� of layer S� situ-
ated in the local AFM surroundings is higher than
temperature Tc of layer S situated in the local FM sur-
roundings. It is physically transparently because the
partial compensation of paramagnetic effect of ex-
change field I takes place only for the S� layer.
Thus in only such nonsymmetrical case the decoup-
led superconductivity and the Tc difference are possi-
ble for the F/S pentalayer. This effect could be ob-
served in special prepared samples in field-cooled
FMAFM configuration.
In common case this FMAFM configuration is ener-
getically less favorable than symmetrical AFM case,
so that this pentalayer can have only AFMS state (see
Fig. 4,d). Comparing corresponding diagrams for the
tetralayer (Fig. 3) and pentalayer (Fig. 4) cases we
conclude that the F/S/F�/S� tetralayer has more rich
properties in physical sense. It proves that this F/S
system is the most perspective candidate for using in
the superconducting nanoelectronics.
Note in conclusion, the details of the phase diagram
significantly depend on the choice of the system pa-
rameters and the above analysis was carried out as-
suming the absence of an external magnetic field
( )H
0 .
4. Conclusions
The few-layered F/S systems have been consis-
tently studied within the modern theory of the prox-
imity effect with a detailed account of the boundary
conditions given. Theoretical studies of the critical
temperature dependence on the thicknesses of the F
layers have been performed as well for a wide range of
parameters, and a physically interesting range of their
values has been determined. The latter should be of
help in choosing materials and technology for prepara-
tion of the F/S systems with predetermined pro-
perties.
It has been shown that when the � phase supercon-
ductivity coexists with the nonequivalence of all lay-
Adjustment of superconductivity and ferromagnetism in the few-layered nanostructures
Fizika Nizkikh Temperatur, 2006, v. 32, No. 8/9 1075
a b
c d
Fig. 4. The phase diagrams of the F/S/F�/S�/F� �
pentalayer with the same parameters as in Figs. 2, 3. All
denotations of curves and states correspond to ones en-
tered in the last part of Sec. 2. The phase diagram for the
FM configuration of the magnetizations of all F layers
(a); the phase diagram for the AFM configuration (b); the
phase diagram for the FMAFM configuration (c); the ge-
neralized phase diagram of the pentalayer (the designa-
tions follow to ones used in Fig. 3,d) (d).
ers the physics of the F/S tetralayer is considerably
richer in comparison with one for the earlier studied
F/S/F trilayer [22,23], the F/S superlattices
[1,31,32], and even the F/S pentalayer studied here.
For tetra- and pentalayer the new � phase supercon-
ducting states have been found. The hierarchy of criti-
cal temperatures is manifested mainly through the oc-
currence of the different critical temperatures in the
different S and S� layers (decoupled superconductiv-
ity). This prediction can be experimentally verified
both for the common case and for the specially pre-
pared field-cooled samples.
Theoretical studies performed in this paper have
shown that the four-layered F/S/F�/S� system has
the best prospects for its use in superconducting spin
electronics (superconducting spintronics). This sys-
tem can be used for a creation of an essentially new
type of nanoelectronics combining within the same
layered sample the advantages of the superconducting
and magnetic channels of data recording that are asso-
ciated with the conducting properties of both S and S�
layers and the magnetic ordering of the
magnetizations of the ferromagnetic layers as it is
shown for the F/S superlattice case [1,35]. Prelimi-
nary estimates show that the possible spin device
based on the studied F/S tetralayer can have up to
seven different states, and transitions between these
states can be controlled by a external magnetic field.
Certainly, low temperatures at which usual «cold» su-
perconductivity is possible would be a condition for
the use of this type of control device. However, simi-
lar superconducting devices on the basis of the F/S
structures with S layers made out of high-temperature
superconducting materials [36] should work at much
higher temperatures.
Y.P. and M.K. are grateful to the Max-Planck-In-
stitute for the Physics of Complex Systems (Dresden)
for providing excellent conditions for fruitful work.
Y.P. thanks Prof. Fulde and the participants of his
seminar for the productive discussion of the results
presented in this paper. This work was supported in
part by RFBR (No 04-02-16761, No 05-02-16369).
1. Y.A. Izyumov, Y.N. Proshin, and M.G. Khusainov,
Usp. Fiz. Nauk 172, 113 (2002); Physics-Uspekhi 45,
109 (2002).
2. V.V. Ryazanov, V.A. Oboznov, A.S. Procofiev, V.V.
Bolginov, and A.K. Feofanov, J. Low Temp. Phys.
136, 385 (2004).
3. A.A. Golubov, M.Y. Kupriyanov, and E. Il’ichev, Rev.
Mod. Phys. 76, 411 (2004).
4. A. Buzdin, Rev. Mod. Phys. 77, 935 (2005).
5. A.I. Buzdin, B. Vuji�i�, and M.Y. Kupriyanov, Zh.
Eksp. Teor. Fiz. 101, 231 (1992) [Sov. Phys. JETP
74, 124 (1992)].
6. Z. Radovi�, M. Ledvij, L. Dobrosavljevi�-Gruji�, A.I.
Buzdin, and J.R. Clem, Phys. Rev. B44, 759 (1991).
7. L.N. Bulaevskii, V. Kuzii, and A. Sobyanin, Pis’ma
Zh. Eksp. Teor. Fiz. 25, 314 (1977) [JETP Lett. 25,
290 (1977)].
8. M.G. Khusainov and Y.N. Proshin, Phys. Rev. B56,
14283 (1997); see also Phys. Rev. B62, 6832 (2000).
9. Y.N. Proshin and M.G. Khusainov, Pis’ma Zh. Eksp.
Teor. Fiz. 66, 527 (1997) [JETP Lett. 66, 562
(1997)].
10. Y.N. Proshin and M.G. Khusainov, Zh. Eksp. Teor.
Fiz. 113, 1708 (1998); see also Zh. Eksp. Teor. Fiz.
116, 1882 (1999); JETP 86, 930 (1998); ibid. 89,
1021 (1999).
11. J. Aarts, J.M. E. Geers, E. Br�ck, A.A. Golubov, and
R. Coehoorn, Phys. Rev. B56, 2779 (1997).
12. L.R. Tagirov, Physica C307, 145 (1998).
13. Y.V. Fominov, N.M. Chtchelkatchev, and A.A. Go-
lubov, Pis’ma Zh. Eksp. Teor. Fiz. 74, 101 (2001)
[JETP Lett. 74, 96 (2001)].
14. Y.V. Fominov, N.M. Chtchelkatchev, and A.A. Go-
lubov, Phys. Rev. B66, 014507 (2002).
15. A. Bagrets, C. Lacroix, and A. Vedyayev, Phys. Rev.
B68, 054532 (2003).
16. Y.A. Izyumov, Y.N. Proshin, and M.G. Khusainov,
Pis’ma Zh. Eksp. Teor. Fiz. 71, 202 (2000) [JETP
Lett. 71, 138 (2000)].
17. I.A. Garifullin, D.A. Tikhonov, N.N. Garif’yanov, L.
Lazar, Y.V. Goryunov, S.Y. Khlebnikov, L.R. Tagi-
rov, K. Westerholt, and H. Zabel, Phys. Rev. B66,
020505 (2002).
18. A.I. Larkin and Y.N. Ovchinnikov, Zh. Eksp. Teor.
Fiz. 47, 1136 (1964) [Sov. Phys. JETP 20, 762
(1965)].
19. P. Fulde and R.A. Ferrell, Phys. Rev. A135, 550
(1964).
20. E.A. Demler, G.B. Arnold, and M.R. Beasley, Phys.
Rev. B55, 15174 (1997).
21. M.G. Khusainov and Y.N. Proshin, Usp. Fiz. Nauk
173, 1385 (2003) [Phys. Usp. 46, 1311 (2003)].
22. A.I. Buzdin, A.V. Vedyayev, and N.V. Ryzhanova,
Europhys. Lett. 48, 686 (1999).
23. L.R. Tagirov, Phys. Rev. Lett. 83, 2058 (1999).
24. F.S. Bergeret, K.B. Efetov, and A.I. Larkin, Phys.
Rev. B62, 11872 (2000).
25. F.S. Bergeret, A.F. Volkov, and K.B. Efetov, Phys.
Rev. B69, 174504 (2004).
26. V. Krivoruchko and V. Varyukhin, Ukr. J. Phys. 50,
795 (2005).
27. F.S. Bergeret, A.F. Volkov, and K.B. Efetov, Phys.
Rev. Lett. 86, 4096 (2001).
28. F.S. Bergeret, A.F. Volkov, and K.B. Efetov, Phys.
Rev. B68, 064513 (2003).
29. A.F. Volkov, F.S. Bergeret, and K.B. Efetov, Phys.
Rev. Lett. 90, 117006 (2003).
30. Y.V. Fominov, A.A. Golubov, and M.Y. Kupriyanov,
Pis’ma Zh. Eksp. Teor. Fiz. 77, 609 (2003) [JETP
Lett. 77, 510 (2003).
1076 Fizika Nizkikh Temperatur, 2006, v. 32, No. 8/9
Y.A. Izyumov, M.G. Khusainov, and Y.N. Proshin
31. M.G. Khusainov, Y.A. Izyumov, and Y.N. Proshin,
Pis’ma Zh. Eksp. Teor. Fiz. 73, 386 (2001) [JETP
Letters 73, 344 (2001).
32. Y.N. Proshin, Y.A. Izyumov, and M.G. Khusainov,
Phys. Rev. B64, 064522 (2001).
33. J.P. Goff, P.P. Deen, R.C. C. Ward, M.R. Wells, S.
Langridge, R. Dalgleish, S. Foster, and S. Gordeev,
JMMM 240, 592 (2002).
34. S. Oh, D. Youm, and M.R. Beasley, Appl. Phys. Lett.
71, 2376 (1997).
35. Y. Proshin, Y. Izyumov, and M.G. Khusainov, Super-
cond. Sci. Techn. 15, 285 (2002).
36. H.-U. Habermeier and G. Cristiani, Phys. Status So-
lidi A201, 1436 (2004).
Adjustment of superconductivity and ferromagnetism in the few-layered nanostructures
Fizika Nizkikh Temperatur, 2006, v. 32, No. 8/9 1077
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