Vortex molecules in thin films of layered superconductors
The seminal work of A.A. Abrikosov on the vortex lattices in type-II superconductors has triggered the creation and rapid development of the whole new field of condensed matter physics, namely, the physics of vortex matter in a large variety of superfluid and superconducting systems. Among these sys...
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Samokhvalov, A.V. Mel’nikov, A.S. Buzdin, A.I. 2021-02-03T19:05:23Z 2021-02-03T19:05:23Z 2018 Vortex molecules in thin films of layered superconductors / A.V. Samokhvalov, A.S. Mel’nikov, A.I. Buzdin // Физика низких температур. — 2018. — Т. 44, № 6. — С. 691-700. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 74.25.Hа, 74.25.Uv, 74.78.–w https://nasplib.isofts.kiev.ua/handle/123456789/176151 The seminal work of A.A. Abrikosov on the vortex lattices in type-II superconductors has triggered the creation and rapid development of the whole new field of condensed matter physics, namely, the physics of vortex matter in a large variety of superfluid and superconducting systems. Among these systems and compounds the layered superconductors are known to form a very important class which is associated with a number of fascinating new discoveries in vortex physics. The phenomenon of the vortex attraction in tilted magnetic fields provides one of the examples of such findings which affected theoretical and experimental research in the field for almost two decades. In our paper we review some recent advances in this direction focusing on the intervortex interaction and equilibrium vortex structures in thin films of layered superconductors in magnetic field tilted with respect to the layers. In such a case the magnetic field penetrates superconductor in the form of tilted vortices or a crossing array of Josephson vortices and pancake stacks. We study the interplay between two different long-range potentials: (i) attraction of tilted vortices or deformed stacks; (ii) the Pearl's repulsion. This interplay is responsible for the formation of the minimum in the total interaction energy and resulting decay of vortex chains in clusters. The number of Abrikosov vortices in these clusters (or vortex molecules) depends on field tilting angle and film thickness. This work was supported, in part, by the Russian Foundation for Basic Research, the EU COST Action CA16218, the French ANR project “Optofluxonics” (AIB) and Foundation for the Advancement of Theoretical Physics “Basis” (ASM). The numerical simulations were carried out with the support of the Russian Science Foundation (project No. 17-12-01383). AIB thanks the Leverhulme Trust for supporting his stay in Cambridge 768 University en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова Vortex molecules in thin films of layered superconductors Article published earlier |
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Vortex molecules in thin films of layered superconductors |
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Vortex molecules in thin films of layered superconductors Samokhvalov, A.V. Mel’nikov, A.S. Buzdin, A.I. Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
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Vortex molecules in thin films of layered superconductors |
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Vortex molecules in thin films of layered superconductors |
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Vortex molecules in thin films of layered superconductors |
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Vortex molecules in thin films of layered superconductors |
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vortex molecules in thin films of layered superconductors |
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Samokhvalov, A.V. Mel’nikov, A.S. Buzdin, A.I. |
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Samokhvalov, A.V. Mel’nikov, A.S. Buzdin, A.I. |
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Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
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Специальный выпуск К 90-летию со дня рождения A.A. Абрикосова |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The seminal work of A.A. Abrikosov on the vortex lattices in type-II superconductors has triggered the creation and rapid development of the whole new field of condensed matter physics, namely, the physics of vortex matter in a large variety of superfluid and superconducting systems. Among these systems and compounds the layered superconductors are known to form a very important class which is associated with a number of fascinating new discoveries in vortex physics. The phenomenon of the vortex attraction in tilted magnetic fields provides one of the examples of such findings which affected theoretical and experimental research in the field for almost two decades. In our paper we review some recent advances in this direction focusing on the intervortex interaction and equilibrium vortex structures in thin films of layered superconductors in magnetic field tilted with respect to the layers. In such a case the magnetic field penetrates superconductor in the form of tilted vortices or a crossing array of Josephson vortices and pancake stacks. We study the interplay between two different long-range potentials: (i) attraction of tilted vortices or deformed stacks; (ii) the Pearl's repulsion. This interplay is responsible for the formation of the minimum in the total interaction energy and resulting decay of vortex chains in clusters. The number of Abrikosov vortices in these clusters (or vortex molecules) depends on field tilting angle and film thickness.
|
| issn |
0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/176151 |
| citation_txt |
Vortex molecules in thin films of layered superconductors / A.V. Samokhvalov, A.S. Mel’nikov, A.I. Buzdin // Физика низких температур. — 2018. — Т. 44, № 6. — С. 691-700. — Бібліогр.: 34 назв. — англ. |
| work_keys_str_mv |
AT samokhvalovav vortexmoleculesinthinfilmsoflayeredsuperconductors AT melnikovas vortexmoleculesinthinfilmsoflayeredsuperconductors AT buzdinai vortexmoleculesinthinfilmsoflayeredsuperconductors |
| first_indexed |
2025-11-26T10:30:19Z |
| last_indexed |
2025-11-26T10:30:19Z |
| _version_ |
1850618726897942528 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6, pp. 691–700
Vortex molecules in thin films of layered superconductors
A.V. Samokhvalov1,2, A.S. Mel’nikov1,2, and A.I. Buzdin3,4
1Institute for Physics of Microstructures, Russian Academy of Sciences, Nizhny Novgorod, GSP-105, 603950, Russia
2N.I. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod 603950, Russia
3University of Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence Cedex, France
4Department of Materials Science and Metallurgy, University of Cambridge, CB3 0FS, Cambridge, United Kingdom
E-mail: samokh@ipmras.ru
Received December 27, 2017, published online April 25, 2018
The seminal work of A.A. Abrikosov on the vortex lattices in type-II superconductors has triggered the crea-
tion and rapid development of the whole new field of condensed matter physics, namely, the physics of vortex
matter in a large variety of superfluid and superconducting systems. Among these systems and compounds the
layered superconductors are known to form a very important class which is associated with a number of fascinat-
ing new discoveries in vortex physics. The phenomenon of the vortex attraction in tilted magnetic fields provides
one of the examples of such findings which affected theoretical and experimental research in the field for almost
two decades. In our paper we review some recent advances in this direction focusing on the intervortex interac-
tion and equilibrium vortex structures in thin films of layered superconductors in magnetic field tilted with re-
spect to the layers. In such a case the magnetic field penetrates superconductor in the form of tilted vortices or a
crossing array of Josephson vortices and pancake stacks. We study the interplay between two different long-
range potentials: (i) attraction of tilted vortices or deformed stacks; (ii) the Pearl's repulsion. This interplay is re-
sponsible for the formation of the minimum in the total interaction energy and resulting decay of vortex chains in
clusters. The number of Abrikosov vortices in these clusters (or vortex molecules) depends on field tilting angle
and film thickness.
PACS: 74.25.Hа Magnetic properties including vortex structures and related phenomena;
74.25.Uv Vortex phases;
74.78.–w Superconducting films and low-dimensional structures.
Keywords: Abrikosov vortices, layered superconductors, vortex chains.
1. Introduction
A.A. Abrikosov contributed to many domains of theo-
retical physics: condensed matter, plasma physics, physics
of high pressure, and quantum electrodynamics. However,
the Nobel Prize in Physics 2003 was awarded to A.A.
Abrikosov for his theoretical discovery of vortices and the
theory of type-II superconductivity. These vortices are now
often called Abrikosov vortices and the understanding of
the properties of the mixed state of type-II superconductors
is ultimately related to the paradigm of “vortex matter”.
Thermal excitations, vortex pinning, crystal anisotropy,
spatial and time dependent magnetic field, all these reveal
panoply of different transitions in the vortex matter, which
makes its physics very rich [1].
Both the equilibrium and transport properties of the vor-
tex matter are essentially affected by the behavior of the
potential of the interaction between vortices. In isotropic
bulk superconductors the intervortex interaction potential is
well known to be always repulsive [2] and screened at
intervortex distances R greater than the London penetration
depth λ. As a result, in perfect crystals quantized Abrikosov
vortices (vortex lines) form a triangular flux lattice [3]. The
magnetic flux through the unit cell of such flux line lattice
equals to the flux quantum 0 = /c eϕ π , and each unit cell is
occupied by the single vortex.
This textbook picture is known to change drastically in
layered superconductors in tilted magnetic field due to the
phenomenon of the long-range attraction between tilted [4–8]
or deformed [9,10] vortex lines composed of two dimension-
al 2D pancake vortices (PVs). For bulk layered superconduc-
tors with a moderate anisotropy (like NbSe2, YBaCuO), the
low magnetic field penetrates mainly in form of the tilted
vortex lines which are arranged in chains (Fig. 1(a)). These
vortex chains have been observed experimentally by the dec-
oration technique in YBa2Cu3O7 [11], scanning-tunneling
© A.V. Samokhvalov, A.S. Mel’nikov, and A.I. Buzdin, 2018
A.V. Samokhvalov, A.S. Mel’nikov, and A.I. Buzdin
microscopy in NbSe2 [12], and Lorentz microscopy meas-
urements in YBa2Cu3O7 [13] (see [14] for a review). For
strongly anisotropic superconductors like Bi2Sr2CaCu2O8+δ
(BSCCO) the chain consists of a lattice of in-plane Josephson
vortices (JVs) crossing with pancake stacks perpendicular to
the planes (Figs. 1(b),(c)). The attraction between PVs and
JVs [15] leads to the deformation of the pancake stacks de-
tected by the modern vortex imaging methods such as Lo-
rentz microscopy [13], high-resolution scanning Hall probe
[17], Bitter decoration techniques [18], and magneto-optical
measurements [19] (see also Ref. 14 as a review). A lot of
other ground-state vortex chain configurations and phase
transitions between them were studied in detail in Ref. 20.
The potential of the interaction between vortices in layered
superconductors depends on the internal structure of vortex
lines and is determined by two different kinds of interac-
tion between pancake vortices in different layers: magnetic
and Josephson interactions. The relative strength of these
two interactions depends on the anisotropy parameter
= /c abΓ λ λ and interlayer spacing s, where cλ is the pen-
etration depth for currents along the c axis and abλ is the
penetration depth for currents in the ab plane (parallel to
the layers). The distinctive feature of anisotropic super-
conductors is known to be an unusual attractive part of the
vortex–vortex interaction potential in tilted magnetic field
[4–10]. Tilted or deformed PVs stacks attract each other in
the plane defined by the anisotropy axes and vortex line, and
the value of the attractive force is controlled by the tilting
angle of the vortex line with respect to the anisotropy axis.
Certainly, the interaction between vortex stacks at small dis-
tances R λ remains repulsive. Interestingly, the attractive
long-range intervortex potential has the same asymptotic
behavior 21/aU R− ( )R λ
for both cases: for tilted
vortex lines [4–8] and for deformed stacks [9,10]. Note, that
the attractive interaction between any two vortices in the
chain (except the nearest neighbors) provides the stability of
the whole vortex chain.
In thin films of layered superconductors, however, such
van der Waals type potential appears to be strongly modified
by the long-ranged Pearl’s repulsion [21] which could mask
the subtle attraction of vortices. The Pearl’s potential ( )PU r
of intervortex interaction is always repulsive, decays ex-
tremely slowly ( 1/PU R
for > )R λ and always domi-
nates at large distances > .mR R λ
It means that the re-
pulsion between two vortices is restored at intervortex dis-
tance > mR R and stability of the vortex chain of a length
mL R is destroyed. As a result, the formation of an infi-
nite vortex chain becomes unfavorable. Adding vortices one
by one we can find an optimal number of vortices which can
be arranged in a chain of a finite length. As a result, there
appears an intriguing possibility to form a vortex structure
consisting of finite size chains, i.e., of vortex molecules. By
varying either the film thickness or the tilting angle we can
modify the balance between the attractive and repulsive in-
teractions, which should determine the size of the energeti-
cally favorable vortex configurations.
Modern vortex imaging techniques provide a possibility
to observe the crossover between different intervortex inter-
action regimes in thin film samples. In particular, the mag-
netic field distributions induced by vortices in thin films can
be probed by the penetrating electron beam used in Lorentz
microscopy measurements [22,23]. This technique, owing to
the low penetration power of the existing 300-kV field-
emission beam, permits us to work with films of thicknesses
smaller than 0.5–1 µm. It is, therefore, “par excellence” an
ideal tool to study the peculiarities of the vortex structures in
thin films. In Ref. 24 it was demonstrated how the Lorentz
microscopy technique permits to discover the special charac-
ters of the intervortex interaction in YBaCuO films of thick-
ness 0.5 m,d µ
placed in a tilted magnetic field. The vor-
tex structure changes qualitatively for a fixed external
magnetic field 0B direction by increasing absolute value of
the field 0B . The increase in the field value 0B causes an
increase in the tilting angles of vortex lines. Therefore, the
vortex attraction prevails at higher 0B values, while at low
fields the attraction force is overcome by the repulsive force
due to Pearl's effect, and the vortex chains are expected to
disappear. Indeed, such a qualitative change in the vortex
structure is confirmed [24] by the experimental data: at low
fields, which correspond to small tilting values, the vortex
chains are completely absent, while at rather high fields
*0 > ,B B where *B is the critical field, the formation of
vortex chains is clearly observed.
Fig. 1. (Color online) Schematic representation of a layered superconductor with tilted (a) and crossing (b), (c) vortex structures, s is a
distance between the layers, N is the number of layers, D = (N – 1)s is the total thickness of the structure.
692 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Vortex molecules in thin films of layered superconductors
In this paper we focus on the calculations of the pair po-
tential of the intervortex interaction in thin films of layered
superconductors. Our calculations are based on the
Lowrence–Doniach theory [25] which is simplified neglect-
ing the effects of weak interlayer Josephson coupling. This
approximation of Josephson-decoupled superconducting
(SC) layers [26] is known to be useful in studies of the vortex
lattice structure for very weak coupling of the layers
= ,J absλ Γ λ
where the Josephson length =J sλ Γ is
the in-plane size of the Josephson core radius [27]. The equi-
librium shape of a single vortex line depends on the in-plane
magnetic field || || 0= BB x distribution and can be very dif-
ferent. Without the in-plane field, the pancakes form a verti-
cal stack. If the in-plane field ||B is rather small
2
|| 0 0( = / )B H sϕ Γ and does not penetrate inside super-
conductor in form of Josephson vortices, then screening
Meissner currents tilt and bend the vortex line. Bending of
the vortex line is essential only near the surface of the SC
film and often can be neglected [28]. In Sec. 2.2 we study the
interaction between two straight stacks of PVs tilted at a cer-
tain angle γ with respect to the c direction, as shown in
Fig. 1(a). In fact in layered superconductors with very weak
interlayer coupling the Josephson vortices will appear at
much lower field. As a result, at tilted magnetic field, cross-
ing lattice of pancakes, formed by Abrikosov vortices, and
Josephson vortices is more energetically preferable than a
lattice of tilted vortex stacks [15–17]. The interaction be-
tween pancakes and in-plane field in the form of Josephson
vortices produces zigzag deformation of the stack of the pan-
cakes [30]. This deformation is responsible for a long-range
attraction between such stacks [9] which is quite similar to
the case of tilted vortex lines. In Sec. 2.3 we study the ex-
treme case when the in-plane magnetic field || 0B H cre-
ates the dense triangular lattice of JVs which are strongly
compressed along the c axis due to a very high anisotropy
1Γ . This dense triangular lattice of JVs produces zigzag
displacement of PVs [30] along the x axis, so that the pan-
cake centers for each vortex line are positioned along the
broken line (see Fig. 1(c)). We calculate the interaction po-
tential between two zigzag lines in a thin SC film. In Sec. 2.4
we compare these two extreme cases and discuss qualitative-
ly the condition of the vortex attraction if the in-plane mag-
netic field is relatively small || 0( )B H and the crossing
points between JVs and PVs are rare. In Sec. 3 we discuss the
condition of vortex cluster formation and phase transitions
between the vortex lattices with different number of vortices
per unit cell.
2. Interaction potential for two stacks of pancakes
As a model of a film of strongly anisotropic supercon-
ductor we consider a finite set of N thin SC layers with
the thickness d much smaller λ located at a distance s
from each other, so that the nth SC layer coincides with the
plane = =nz z ns (1 ).n N≤ ≤ The field component per-
pendicular to the layers creates the vortex line which pierc-
es the film as a stack of PVs centered at the points
0 0=n n nx y+r x y in the nth layer. We derive general ex-
pressions for interaction potential of two identical stacks of
PVs taking into account both long-range attraction and
repulsion phenomena. We study both the limits of weak
|| 0( )B H and strong || 0( )B H in-plane magnetic
field. The shape of the interacting vortex lines is assumed
to be fixed and not affected by the vortex–vortex interac-
tion potential. Certainly, such assumption is valid only in
the limit of rather larger distances between the vortex lines
when the effect of interaction on the vortex shape can be
viewed as a small perturbation.
2.1. Basic equations
Within the model of the set of Josephson-decoupled SC
layers, pancakes interact with each other only via magnetic
fields. General equation for the vector potential A distribu-
tion in such system reads
, =1
4rot rot = ( ) ( )
N
m
n n
n m
z z
c
π
δ −∑A J r . (1)
The sheet current at the nth layer created by the pancake at
mth layer takes the form
( ) = ( ) ( , ) ,
4
m m
n m nm n
c z − δ − πΛ
J r Φ r r A r (2)
where 2 2= / = /abd sΛ λ λ is the effective penetration depth
in a superconducting film, ( , )m zA r is the vector potential
induced by the only pancake vortex located in the mth layer.
The vector ( )Φ r in the Eq. (2) is given by the expression
[ ]00
2( ) =
2
×ϕ
π
z r
Φ r
r
. (3)
For the layered system without Josephson coupling a general
expression for the free energy can be written in the form
( )
2
2 2
=1
1 4= rot ( ) ( ) .
8
N
n n
n
F dV z z
c
π + Λ δ − π
∑∫ A J r (4)
The total vector potential ( , )zA r and the sheet current in
the nth layer ( )nJ r , produced by an arbitrary vortex line
are the sum of the contributions induced by N pancakes:
=1 =1
( , ) = ( , ), ( ) = ( ),
N N
m m
n n
m m
z z∑ ∑A r A r J r J r
and can by found from Eqs. (1)–(3) using an approach simi-
lar to that proposed in Refs. 32, 33 (see Ref. 28 for details).
Using the gauge div = 0A and the Fourier transforms
2( , ) = e ( , ),i ikzk d dz z+∫ qrA q r A r (5)
2 2( ) = e ( , ), ( ) = e ( ),i i
n n n nd z d∫ ∫qr qrA q r A r J q r J r (6)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 693
A.V. Samokhvalov, A.S. Mel’nikov, and A.I. Buzdin
one can rewrite the basic Eq. (1) for each vortex line in the
momentum representation as follows:
[ ]2 2 1( ) ( , ) = ( ) ( ) e ,ikns
n n
n
q k k+ −
Λ ∑A q Φ q A q (7)
where
[ ]0
0 2( ) = ( )e , ( ) = .i nn i
q
×
− ϕqr z q
Φ q Φ q Φ q
These equations can be reduced to the scalar form
| |1 e = e ,
2
iqn m qs nn m
m
f f
q
− −+
Λ ∑ r (8)
where we introduce the new functions ( )nf q :
( )( ) = ( ) ( ) = ( ) ( )
4 4n n n n
c c f−
πΛ πΛ
J q Φ q A q Φ q q . (9)
The solution of the linear system (8) for a fixed configura-
tion of pancakes nr determines the distribution of the vec-
tor potential ( , )zA r which is created by an arbitrary vor-
tex line in a finite stack of superconducting layers.
For two vortex lines we can write the total vector poten-
tial and the total sheet current as superpositions of contri-
butions coming from the first (1)( nA , (1) )nJ and second
(2)( nA , (2) )nJ vortices. Calculating the interaction energy
intε of vortex lines we should keep in the free energy only
the terms which contain the products of fields correspond-
ing to different vortex lines. As a result, in the momentum
representation the general expression (4) for the free ener-
gy of the layered system without Josephson coupling reads
( )2 (1) (1) (2)
int 3
1= ( ) ( ) ( )
32
n n n
n
d ε − − +π Λ
∑∫ q Φ q A q Φ q
( )(2) (2) (1)( ) ( ) ( ) .n n n
+ − − Φ q A q Φ q (10)
For the particular case of two identical (parallel) vortex
lines which are shifted at the vector (2) (1)= ( = 1 )n n n N− −R r r
in the xy plane we get following general expression for the
interaction energy via the scalar functions ( )nf q [28]:
2 2
0
int 3 2( ) = cos( ) ( )e .
16
i nn
n
d f
q
−ϕ
ε
π Λ
∑∫ qrqR qR q (11)
The expression (11) and equations (8) determine the inter-
action energy of two identical vortex lines.
2.2. Interaction potential of tilted stacks of pancakes
In the limit of weak in-plane magnetic field || 0( )B H
we neglect the vortex line bending and restrict ourselves to
the case of straight vortex lines parallel to the plane xy and
tilted at a certain angle γ with respect to the anisotropy axis c
(z axis) as shown in Fig. 1(a). Let us evaluate the interaction
energy (11) of two tilted parallel vortex lines taking
(1) (2) (1)
0= tan , =n n nns γ +r x r r R .
For the case of a large intervortex distance ,R s the
solution of the problem (8) may be simplified, because for
the relevant wave vectors 1/q R
the conditions of the
continuous limit are satisfied: 1qs and tan 1xq s γ .
Introducing a continuous coordinate =z ns and continu-
ous function ( )f zq , the linear system of Eq. (8) reduces to
the following integral equation
/2
tan| |
2
/2
1( ) e ( ) = e .
2
D
iq zq z z x
ab D
f z dz f z
q
γ− − ′
−
+ ′ ′
λ ∫q q (12)
The Eq. (12) can be rewritten as a differential one
( ) ( )
2
tan2 2 2 22
2 ( ) = etan
iq zxab x
d f
q f z q q
dz
γ−− λ + − γ +q
q (13)
at the interval /2 < < /2D z D− with the boundary conditions
tan /2
/2
= ( tan )e ,iq Dxx
D
df
q f iq q
dz
± γ
±
± γ ±
q
q (14)
where = ( 1)D N s− is the thickness of superconducting
film. Skipping the details of the solution of Eqs. (13), (14),
we get the following expression for the interaction energy
(11) in the continuous limit [28]:
2 2
0
int 3 2 2= cos( ) ( ),
16 ab
d S
q
ϕ
ε
π λ ∫
q qR q (15)
where the expression for the kernel ( )S q takes form
____________________________________________________
2 2 2 22 2
2 2 2 2 2 2
2(1 ) (1 )sinh ( )(cosh cos( )) 2 sin( )
( ) = ,
1 (1 ) 2 cosh (1 )sinhab
k k p L k p L pL kp pLp kS D
p q p k L k L−
− − + − − ++ +
+ + λ + + +
q (16)
2 2 2 2 2 2= , = / , = tan / .ab ab x abL D q k q q p q q− − −+ λ + λ γ + λ
_______________________________________________
The first term in the kernel (16) describes the intervortex
interaction in the bulk system, while the second term is
responsible for the Pearl effect due to film boundaries.
The minimum energy configuration corresponds to the
case = 0yR . In Fig. 2 we present some typical plots of the
interaction energy int ( , = 0)x yR Rε vs the distance =xR R
694 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Vortex molecules in thin films of layered superconductors
for = 3 abD λ which corresponds to the Lorentz microscopy
experiments in YBCO [24] and Bi-2212 [13] samples. Ana-
lyzing the dependence int ( ),Rε one can separate three con-
tributions to the energy of vortex–vortex interaction: (i) a
short-range repulsion which decays exponentially with in-
creasing intervortex distance R (for > );abR λ (ii) an
intervortex attraction which is known to be specific for tilted
vortices in anisotropic systems; this attraction energy term
decays as 2R− and strongly depends on the angle γ be-
tween the vortex axis and the c direction; (iii) long-range
(Pearl) repulsion which decays as 1R− and results from the
surface contribution to the energy. Note that the third term
does exist even for a large sample thickness D (see Ref. 29)
although in the limit abD λ
it is certainly masked by the
dominant bulk contribution. At abR λ
the short-range
interaction term vanishes and the interaction energy vs R
takes the simple form
2 2
0 eff
int 2 2
2tan ,
8
D
RR
ϕ γ
ε − + π
(17)
where eff = 2 tanh( /2 )ab abD D D− λ λ is the effective
film thickness. One can observe here an interplay be-
tween the long-range attractive (first term in Eq. (17))
and the long-range repulsive (second term in Eq. (17))
forces. Note that the abλ value increases with an increase
in temperature, thus, the effective thickness decreases and
the long-range attraction force appears to be suppressed
with increasing temperature. For large R the energy is
always positive and corresponds to the vortex repulsion
similar to the one between the pancakes in a single layer
system. With a decrease in the distance R the attraction
force comes into play resulting in the change of the sign
of the energy. Such behavior points to the appearance of
the minimum in the interaction potential.
2.3. Interaction potential of zigzag stacks of pancakes
In the limit of strong in-plane magnetic field || 0( )B H
the tilted magnetic field penetrates in the form of the crossing
lattice comprising the in-plane Josephson vortices and per-
pendicular to the layers stacks of pancakes (see Fig. 1(c)). In
high-field limit, || 0 0> = / ,JB H sϕ λ all interlayer junctions
are homogeneously filled with Josephson vortices, which
form the dense triangular lattice. The period of the lattice in
z direction is equal 2s which is much less than the period
along the y axis 0 ||= / .Ja B sϕ λ The interaction between
pancakes and in-plane field in the form of JVs is known to
produce zigzag deformation of PV stacks (see Fig. 1(c))
[15,20,30]. The zigzag deformation is somewhat larger near
the surface (due to the decrease of the stifness of the PVs
stack) but for simplicity we neglect this relatively small ef-
fect. The amplitude of the zigzag deformation
||
0
| | = , = 1,
ln
Jab
n
ab
B
u u
H
λλ
β
β β λ
(18)
is assumed to be the same for all N layers and corre-
sponds to the equilibrium form of the zigzag vortex in a
bulk ( )N → ∞ layered superconductor [9,30]. In this case
the centers of pancakes for both vortex lines are positioned
at a zigzag line
(1) 1 (2) (1)
0= ( 1) =n
n n nu−− +r x r r R , (19)
and the Eq. (8) with respect to functions ( )nf q looks dif-
ferently for odd and even n . Introducing two new func-
tions 1 ( )kf q and 2 ( )kf q
1
2
, if — odd, = ( 1)/2,
=
, if — even, = /2,
k
n
k
f n k n
f
f n k n
+
the Eq. (8) can be rewrited as follows:
2| |
1,2 1,2
1 e
2
k m qs
k m
m
f f
q
− −+ +
Λ ∑
(2| | 1)
2,1
1 e = e .
2
iq uk m qs xm
m
f
q
±− − ++
Λ ∑ (20)
The interaction energy (11) expressed in terms of the scalar
functions 1 ( )kf q and 2 ( )kf q reads
2 2
0
int 3 2( ) = cos( )
16
d
q
ϕ
ε ×
π Λ ∫
qR qR
1 2( )e ( )e .
N
iq u iq ux xk k
k
f f− × + ∑ q q (21)
The expression (21) and the system of algebraic equation
(20) determine the interaction energy of two identical zigzag
vortex lines which are shifted at the vector R in the xy plane.
Fig. 2. (Color online) Typical plots of the interaction energy per
vortex [Eqs. (15) and (16)] vs the distance R between two vorti-
ces for a film of thickness D = 3λab and different tilting angles
γ = 70°, 75°, 78°, 80° 2 3
0 0( = /16 )abε ϕ π λ .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 695
A.V. Samokhvalov, A.S. Mel’nikov, and A.I. Buzdin
For the case of a large intervortex distance , ,R s u
we may use the continuous limit for the solution of the
problem (20) because for the relevant wave vectors
1qs and 1xq u . We introduce a continuous coordi-
nate = 2z ks and continuous functions 1,2 ( , )f zq defined
at the interval | | /2z D≤ , where D is the total thickness
of superconducting film. Thus, the linear system of
Eqs. (20) reduces to the following integral equations:
/2
| |
1,2 1,2
/2
1( , ) e ( , )
4
D
q z z
D
f q z dz f q z
q s
− − ′
−
+ +′ ′
Λ ∫
/2
| |
2,1
/2
e e ( , ) = e .
4
Dqs
iq uq z z x
D
dz f q z
q s
−
±− − ′
−
+ ′ ′
Λ ∫ (22)
Using the solutions of Eqs. (22), we may rewrite the ex-
pression for interaction energy (21) as follows (see Ref. 34
for details):
____________________________________________________
2 2 2 2
20
int 3 2 2 2 4
( ) ( ) 2 ( )cos sin cos( ) = cos( ) ,
16 (1 / tanh )
x x x
ab
q u q u q q u
d D
s p q p q p L+ + + +
ϕ ε + +
π Λ λ +
∫R q qR (23)
where
2 2 2 2 21/ , , = /2.abp q p q L p D+ − ± ±+ λ (24)
Performing in (23) the necessary integration, we finally obtain the following expression for the interaction energy of two
identical zigzag pancake stacks which are shifted at the vector 0= RR x in the xy plane:
2 2 2
0
int 0 0 02 2
2 2 4( ) = 2 ln
32 ab ab ab
R R u R u R uR D K K K
s R
ϕ − + −ε + + + + λ λ λ π Λ
0 0 0
2 2 2 2 2 2 2 21
0
2 ( ) ( ( 2 )) ( ( 2 ))2 ,
( ) 1 / ( /2)tanhab ab ab ab
J qR J q R u J q R udq
q q Dq q
∞
− − −−
+ − + + +
λ + λ + + λ + λ
∫ (25)
_______________________________________________
where 0J and 0K are the Bessel and modified Bessel
functions of zero order, respectively. The first term propor-
tional to D in Eq. (25) describes the interaction between
the zigzag stacks in the bulk layered system [9], while the
last term is responsible for the influence of film bounda-
ries. In Fig. 3 we present some typical plots of the interac-
tion energy per one layer int /Nε vs the intervortex dis-
tance R for different thickness of the film D and the in-
plane magnetic field 0=xB H .
At long distances abR λ
, the modified Bessel func-
tions decay exponentially, and a bulk short-range repulsion
between two PVs stacks is negligible small. The leading
bulk contribution comes from the logarithmic term in (25),
which describes an attraction between the zigzag PV lines.
The last term in (25) describes long-range Pearl's repulsion
which decays as 1/R and results from the surface contribu-
tion to the energy. Assuming that the effect of the zigzag
deformation abu λ
is small, the long-range part of inter-
action energy (25) for abR λ
can be written as
2 2 2
0
int 2 2 2 3
2 4( ) .
8 ab
Du uR
RR R
ϕ
ε − + +
π λ
(26)
One can observe an interplay of the long-range attractive
and the repulsive forces between two zigzag deformed PV
stacks, similar to the one between two tilted vortices in ani-
sotropic SC films [24,28]. The last term in (26) describes a
Fig. 3. (Color online) Typical plots of the interaction energy per
layer εint(R)/2Nε0 [Eq. (25)] vs the distance R between two zigzag
deformed stacks of pancakes for in-plane magnetic field B||/H0 = 1.
The numbers near the curves denote the values of film thickness
D/λab. The dashed line shows the interaction energy between two
zigzag vortices in bulk (D → ∞) layered SC. Dotted lines show the
long-range part of interaction energy (26). Here 2 2
0 0= /32 ,abε ϕ π λ
s = 0.01λab, γ = 300.
696 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Vortex molecules in thin films of layered superconductors
modification of long-range Pearl’s repulsion due to zigzag
deformation of the PVs stack. Certainly, in a bulk sample
( )D → ∞ the Pearl’s term in (26) vanishes, and at long dis-
tances the dominant interaction between the zigzag PV lines
is an attraction [9]. As a result, in a bulk sample there always
exists the minimum of the interaction potential, which reali-
zes with logarithmic accuracy at 2 ln( / )m ab abR u≈ λ λ .
However the second term in (26) is very important even for
a rather thick films: for large R the energy intε is always
positive and corresponds to the vortex repulsion. With a
decrease of the distance R the attraction force comes into
play and can result in the change of the sign of the energy at
2 2
0 /2 ,ab abR Du≈ λ λ while short-range repulsion is still
weak. Such behavior should be accompanied by the appear-
ance of the minimum in the interaction potential. The condi-
tion 0 > mR R gives us a rough estimate of the critical film
thickness crD for which the minimum in the interaction
potential can exist and the formation of vortex chains can be
energetically favorable:
22
||2 3
cr
0
4
4 (ln ) ,J
ab
ab
B
D
H
σλ
≈ σλ β β λ
(27)
where the constant σ is of the order unity. The more accu-
rate criterion of attraction of zigzag PVs stacks should be
based on the expression for the interaction energy (25) and
may be found from the conditions
int int( ) = ( ) = 0.R Rε ε′
The typical dependence of the critical film thickness crD
as a function of in-plane magnetic field ||B is shown in
Fig. 4. So, in the presence of a dense lattice of Josephson
vortices, the AVs penetrate in the form of chains only for a
large film thickness cr>D D . Otherwise, if cr<D D , the
formation of the usual Abrikosov lattice of zigzag de-
formed stacks of PVs occurs. It is interesting to note that
following (26) the intervortex attraction increases near the
critical temperature of the superconducting transition ,cT
when abλ becomes large (except for the region very close
to cT where the condition ab Jλ λ
is not satisfied). This
behavior is in contrast with the moderately anisotropic case
[28] when the critical thickness for intervortex attraction to
be observed decreases near cT and the repulsion between
vortices prevails.
2.4. Interaction potential of two stacks in the crossing
lattices
In the high in-plane field limit || 0B H considered in
Sec. 2.3 the triangular lattice of JVs is so dense that the cur-
rents of adjacent JVs ovrlap setrongly and the amplitude of
the zigzag deformation decreases [30]. As a result, the criti-
cal thickness crD grows rapidly with an increase of the in-
plane field ||B (see Fig. 4). Also the potential dip in the
intervortex interaction energy (25) strongly decreases with
an increase of ||B [9]. Optimal regime for the long-ranged
attraction corresponds to || 0B H . If the magnetic field ||B
is small, the crossings between JVs and PVs are rare and the
distance between Josephson vortices along the z axis be-
comes much larger than 2s, as well as the distance between
the deformed parts of the PVs line (see Fig. 1(b)). Let us
estimate the attraction between two deformed stacks in this
limit. The Josephson vortices are well separated and posi-
tioned at the distance
0 || 0 ||3 /2 /s sB H Bγϕ
along the z axis. As has been demonstrated in Ref. 9, the
contribution to the attraction from one crossing is
22
0
att 2( ) = ,
ln( / )
ab
J J ab
s
R
AR
ϕ λ
ε − λ λ λ
(28)
where 3.5A ≈ (see Ref. 15). Neglecting the effect of a
zigzag deformation on the Pearl's repulsion, the long-
ranged part of interaction energy int ( )Rε in low-field limit
|| 0B H at distances abR λ
may be estimated as
2 2
0
int 2 2 2
2( )
8 ab
DuR
RR
ϕ
ε − +
π λ
, (29)
where the effective displacement [9,15]
1/42
||
0
2 2
ln( / )
ab
J J ab
B
u
A H
λ
λ λ λ
(30)
grows slowly with increasing in-plane magnetic field ||.B
The expressions (29), (30) result in the following estimate
of the critical film thickness cr :D
1/22 2 ||
cr
0
( / )ln .
2
J J ab
ab
BA
D
H
− λ λ λ
λ
(31)
Fig. 4. Critical film thickness Dcr as a function of in-plane mag-
netic field B|| ( ).− −• The dashed line shows the estimate of the
critical film thickness (27). Here β = B||λJ/H0λab, s = 0.01λab, γ =
= 300, λJ = 3λab, σ = 2.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6 697
A.V. Samokhvalov, A.S. Mel’nikov, and A.I. Buzdin
As before, the case || 0B H is optimal for observation of
intervortex attraction and this case corresponds to the
crossover between the regimes described by formulas (27)
and (31). For typical experimental value of the in-plane
magnetic field [17] B|| = 20–30 G we obtain cr 70 μmD .
Thus, the relations (27), (31) give us the estimate of the
threshold value of the thickness
cr || 0
cr || 0
, if < ,
>
, if > ,
D B H
D
D B H
for which the minimum in the interaction potential can
exist and the interplay between the long-range attraction
and repulsion between zigzag vortex stacks lines in the
films of layered superconductors takes place. We should
note that the Pearl's interaction plays an important role
even for rather thick film. Indeed, for the most favorable
case || 0B H this interaction completely masks the attrac-
tion for the film thicknesses smaller than ∼50–200 λab.
3. Vortex molecules
The vortex attraction in bulk layered superconductors is
known to result in the formation of infinite vortex chains.
But even in the regime when the intervortex attraction ex-
ists, the formation of infinite chains can be questioned for
rather thin films. The point is that, despite the fact that two
vortices attract each other, further increase in the number of
vortices arranged in a chain can be energetically unfavorable
because of the slower decay of the long-ranged Pearl's re-
pulsive force compared to the attractive one. Therefore, for
rather thin samples, there appears an intriguing possibility to
observe vortex chains of finite length, i.e., vortex molecules
or clusters. The calculations [24,28] confirm this and indeed
the number of vortices energetically favorable in a molecule
grows as we increase the film thickness and/or the tilting
angle because of the increasing attraction term in the pair
potential intε (15), (16). In Ref. 28 the intervortex interac-
tion has been also calculated and analyzed taking account of
the equilibrium shape of tilted vortex lines.
Figure 5 shows schematic pictures of vortex matter con-
sisting of dimeric (Fig. 5(a)) and trimeric (Fig. 5(b)) mole-
cules, indicating that the vortex lattices contain more than
one vortex per unit cell. The transitions between different
multiquanta vortex lattices should occur with the change of
the tilting angle and field strength [28]. Finally, for rather
thick samples with abd λ
we get a standard infinite chain
structure typical of bulk systems (Fig. 5(c)). Note that the
formation of an infinite vortex chain may be considered in
some sense as a polymerization of the vortex molecules.
Certainly, the crossover from the vortex molecule state to
the infinite chain structure is strongly influenced by the in-
crease in the vortex concentration governed by the compo-
nent of the external magnetic field perpendicular to the film.
Indeed, one can expect such a crossover to occur when the
mean intervortex spacing approaches the molecular size.
The experiments [24] were performed at constant orien-
tation of the applied magnetic field. Therefore, by varying
the magnetic field, changes of both the tilting angle and vor-
tex concentration were realized. As the vortex concentration
was relatively high, we could not expect to observe the mol-
ecules (in this regime the average distance between vortices
must be much larger than the size of the molecule). To ob-
serve the vortex molecules (or multiquanta vortex lattices) it
would be preferable to change only the parallel component
of the magnetic field, by varying the vortex tilting angle
while not affecting the vortex concentration, which must be
very low to avoid the inter-molecule interaction.
Qualitatively all conclusions of references [24,28] are ap-
plicable for the case of crossing lattices. Similarly, the zig-
zags of PVs stacks can be arranged in vortex molecules. To
estimate the size of such molecule let's find the cohesion en-
ergy of equidistant chain of deformed PVs stacks. The per-
pendicular component of the magnetic field B⊥ fixes the
number M of AVs in the film, so that in the square film
W W× it should be 2
0= /M B W⊥ ϕ vortices. We consider
the case of small concentration of vortices when at large dis-
tances the intervortex interaction is given by Pearl’s term. For
roughly uniform vortex distribution (usual vortex lattice) the
energy per one vortex can be estimated as 2
0 0 /4 .B W⊥ε ϕ π
If vortices form clusters with m vortices each, it may be
demonstrated that interaction between clusters gives the same
contribution 0ε per one vortex. The remaining energy origi-
Fig. 5. Schematic pictures of vortex matter consisting of dimeric molecules (a), trimeric molecules (b) and infinite vortex chain (c). Vor-
tex positions are denoted by filled ellipses.
698 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
Vortex molecules in thin films of layered superconductors
nates from the interaction between vortices inside cluster, and
for the case of a low in-plane magnetic field || 0 ,B H it
may be expressed via the long-ranged part of interaction en-
ergy intε (29):
int int
>
1= ( ),
m
m
ij
i j
R
m
ε ε∑ (32)
where ijR are the distances between ith and jth vortices in
the chain molecule. Taking the characteristic distance be-
tween vortices in a cluster 2 ln( / )m ab J abR ≈ λ λ λ we find
2 2
0
int 2 2 2
2 ln .
8
m
mab m
Du m
RR
ϕ
ε − +
π λ
(33)
Certainly the cluster formation occurs if int < 0mε and the
number of vortices in the cluster is given by the expression
2
2exp
2 ab m
Dum
R
≈
λ
. (34)
The condition > 1m 2 2( /2 > 1)ab mDu Rλ
gives us the low
boundary of the in-plane magnetic field restricting the in-
terval of vortex molecules existence
22 3||
0
( / )ln> .
2
J J ab
ab
B
H D
λ λ λ
λ
(35)
For = 100 abD λ , = 3J abλ λ and 0 = 3 TH we obtain
|| > 60 GB . Note that the number of vortices m in the
molecules rapidly increases as the in-plane magnetic field
||B grows.
Naturally this scenario realizes only for low concentra-
tion of AVs when the average distance between vortices
0 /B⊥ϕ is much larger than the intervortex distance in the
chain mR . In the opposite case we may expect the existence
of multiquanta flux lattice with several vortices per cell
similar to the case of tilted vortices considered in Ref. 28.
4. Conclusions
To sum up, we have analyzed some distinctive features
of the vortex matter in thin films of layered superconduc-
tors focusing mainly on the effects caused by the tilted
magnetic field. The interplay of electrodynamic interac-
tions inside and outside the film results in a very peculiar
behavior of the intervortex interaction potential which in-
cludes both attractive and repulsive terms. This interplay
can change essentially the structural properties of the vor-
tex matter which become quite different from the ones
suggested in the original work by A.A. Abrikosov. In par-
ticular, we predict the possible formation of vortex clusters
or molecules which can be also viewed as the multiplica-
tion of the unit cell for a regular vortex array.
This work was supported, in part, by the Russian
Foundation for Basic Research, the EU COST Action
CA16218, the French ANR project “Optofluxonics”
(AIB) and Foundation for the Advancement of Theoreti-
cal Physics “Basis” (ASM). The numerical simulations
were carried out with the support of the Russian Science
Foundation (project No. 17-12-01383). AIB thanks the
Leverhulme Trust for supporting his stay in Cambridge
768 University.
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700 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 6
https://doi.org/10.1038/360051a0
https://doi.org/10.1103/PhysRevB.79.094510
https://doi.org/10.1051/jphys:0199000510170197100
https://doi.org/10.1103/PhysRevB.43.7837
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https://doi.org/10.1103/PhysRevB.44.10234
https://doi.org/10.1103/PhysRevB.82.104511
https://doi.org/10.1063/1.1707989
https://doi.org/10.1103/PhysRevB.53.6634
https://doi.org/10.1103/PhysRevLett.86.886
https://doi.org/10.1103/PhysRevB.66.132505
https://doi.org/10.1103/PhysRevLett.97.237005
https://doi.org/10.1103/PhysRevLett.97.237005
https://doi.org/10.1016/0921-4534(93)90497-E
https://doi.org/10.1103/PhysRevB.55.6636
1. Introduction
2. Interaction potential for two stacks of pancakes
2.1. Basic equations
2.2. Interaction potential of tilted stacks of pancakes
2.3. Interaction potential of zigzag stacks of pancakes
2.4. Interaction potential of two stacks in the crossing lattices
3. Vortex molecules
4. Conclusions
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