APPLIED PHYSICS LETTERS
VOLUME 76, NUMBER 18
1 MAY 2000
Magnetic pinning in superconductor-ferromagnet multilayers
L. N. Bulaevskiia)
Department of Physics and Astronomy, CUNY Lehman College 250 Bedford Park Boulevard West, Bronx,
New York 10468-1589; Los Alamos National Laboratory, Los Alamos, New Mexico 87545
E. M. Chudnovsky
Department of Physics and Astronomy, CUNY Lehman College, 250 Bedford Park Boulevard West, Bronx,
New York 10468-1589
M. P. Maley
Los Alamos National Laboratory, Los Alamos, New Mexico 87545
共Received 18 January 2000; accepted for publication 9 March 2000兲
We argue that superconductor/ferromagnet multilayers of nanoscale period should exhibit strong
pinning of vortices by the magnetic domain structure in magnetic fields below the coercive field
when ferromagnetic layers exhibit strong perpendicular magnetic anisotropy. The estimated
maximum magnetic pinning energy for single vortex in such a system is about 100 times larger than
the pinning energy by columnar defects. This pinning energy may provide critical currents as high
as 106 ⫺107 A/cm2 at high temperatures 共but not very close to T c 兲 at least in magnetic fields below
0.1 T. © 2000 American Institute of Physics. 关S0003-6951共00兲02618-8兴
Technological applications of superconductors depend
crucially on the value of the critical current, which is determined by the pinning of vortices. Enhancement of pinning
has been the major goal in developing new superconducting
materials. The problem of pinning is especially important for
high temperature superconductors 共HTSC兲 at liquid nitrogen
temperatures. At that temperature typical pinning potentials
are not high enough to prevent depinning by thermal fluctuations, especially because of the small size of the vortex core
and the layered structure of copper oxides.
In the past the enhancement of the pinning in conventional superconductors was done through manufacturing
samples with microscopic defects. Several new approaches
have been developed in recent years. They include macroscopic modulation of the pinning potential via, e.g., modulating the thickness of a superconducting film,1 manufacturing films with micrometer-scale holes,2 or putting magnetic
particles3 or magnetic dots on the surface of superconducting
films.4 For HTSC materials a remarkable effect has been
achieved in samples with columnar defects produced by
heavy ion irradiation.5,6 All the above methods are based
upon introducing defects that suppress superconductivity.
The pinning then arises from the tendency of the vortex normal core to match with the region where the superconductivity is suppressed. The maximal energy per unit length of the
vortex, available for nonmagnetic pinning, is the condensation energy of Cooper pairs in the volume of the vortex core,
U cp ⬃(H 2c /8␲ ) ␲␰ 2 ⬇(⌽ 0 /8␲ ␭ L ) 2 , where H c is the thermodynamic critical field, ␰ is the radius of the core 共the superconducting coherence length兲, ⌽ 0 is the flux quantum, and
␭ L is the London penetration length, ␭ L ⬃0.15 ␮ m for
HTSC. At low temperature U cp ⬃1000 K per crystallographic unit cell for columnar defects along the c axis. It
drops fast as T approaches T c because of the increase of the
London penetration length.
a兲
Electronic mail: lnb@viking.lanl.gov
In this letter we propose to pin the magnetic flux of
vortices rather than their cores. Such magnetic pinning can
be achieved in superconducting-ferromagnetic 共SC/FM兲 multilayers separated by a buffer layer to avoid chemical interaction between the layers, Fig. 1. We assume here that FM
layers have strong coercivity. Let us consider FM layers with
strong perpendicular magnetic anisotropy. In the external
perpendicular magnetic field H, below the coercivity field
H coer , they have stripe-like domain structures shown in Fig.
1. In the following we shall assume a regular stripe structure
with the same width of domains, l, in all layers. The effect of
irregularities will be discussed later. Applied to HTSC, the
SC layers are parallel to the CuO planes. We shall assume
that the thickness of the SC layers, d s , is small in comparison with l and with the London penetration length, ␭ L Ⰶl.
The thickness of the FM layers, d m , is assumed small compared with d s . In this case the SC system is practically threedimensional and the effective London penetration length,
␭ L 关 (d s ⫹d m )/d s 兴 1/2⬇␭ L . The stripe domain structure in the
FM layers modulates the magnetic field through the SC layers creating a pinning potential
FIG. 1. Structure of ferromagnet/superconductor multilayers. Inset shows
magnetic fields in a periodic stripe-like domain structure with the magnetization perpendicular to the layers.
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© 2000 American Institute of Physics
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Appl. Phys. Lett., Vol. 76, No. 18, 1 May 2000
U mp 共 x 兲 ⬃⌽ 0 M 共 x 兲 d s
Bulaevskii, Chudnovsky, and Maley
2595
共1兲
for a single vortex line, where M (x) is the magnetization of
FM domain. Ignoring the width of the domain wall, M (x)
⫽⫾M 0 .
An estimate of the amplitude of the one-vortex pinning
potential, ⌽ 0 M 0 d s , follows from the Gibbs energy,
n⌽ 0 H m /4␲ , of vortices with concentration n in the local
magnetic field of the FM film, H m ⫽4 ␲ M . It can be understood as the energy of the attraction between the vortex and
the ferromagnetic domains. One can also obtain this result by
considering the interaction of an extra vortex, created by an
external field, with the structure shown in Fig. 1. The energy
of the extra vortex would oscillate in space, creating a pinning potential of the amplitude found above.
FM films of orthoferrites, Tb–Fe, Co–Pt, and others can
be used.7,8 In Tb–Fe and Co–Pt films the perpendicular anisotropy is independent of the film thickness, d m , from 500
Å up to 1 ␮m with domain widths in the micrometer range.7,8
For Tb–Fe and Co–Pt films studied to date the coercive field
is of order of a few kG. When the magnetic field exceeds the
coercive field, domains of the magnetization opposite to the
field begin to shrink and the pinning of vortices disappears.
Below the coercive field the domain structure in the FM
layers is frozen. For M 0 ⫽500 emu the elemental magnetic
single vortex pinning energy is about 105 K, that is one hundred times the energy of the pinning by columnar defects. If
␭ L does not exceed d s and l, the magnetic pinning depends
on temperature through M 0 only. In HTSC materials U mp
will be practically independent of temperature for ferromagnets whose Curie temperature is higher than room temperature. On the contrary, any pinning by structural defects decreases with temperature as ␭ L⫺2 (T)⬀(T c ⫺T). The high
value of U mp and its weak temperature dependence is especially important in highly anisotropic Bi- and Tl-based
HTSC where thermal fluctuations destroy pinning at a rate
proportional to exp(⫺Up /T).
Notice that the FM/SC multilayered system with buffer
layers between the FM and the SC layers differs from the
intensively studied SC film with magnetic dots technologically, but the physics of the pinning effect may be not very
different. Magnetic dots may induce alternating magnetic
field 共concentrate magnetic flux in or around the magnetic
material兲 similar to that induced by domains in the multilayer
system and they may also suppress superconductivity in the
region around the dots due to the proximity effect which is
especially strong on the ferromagnet-superconductor
boundary.4 This suppression gives additional pinning but
suppresses the current carrying ability of the superconductor.
The relative role of these mechanisms in films with magnetic
dots is yet unclear. In FM/SC multilayers with buffer layer
between MF and SC layers the proximity effect is negligible
and effect of magnetic field induced by domains is well controlled.
The critical current in the vortex state in the presence of
a one-dimensional pinning potential was discussed previously in connection with superconducting films of periodically modulated thickness.1,9,10 Description of pinning in
such a system is similar to that in FM/SC multilayer in the
external fields H⭓4 ␲ M 0 . At H⬍4 ␲ M 0 the effective magnetic field in SC, B(x)⫽H⫹4 ␲ M (x), changes sign in
FIG. 2. Fundamental matching configurations of a triangular vortex lattice
in a one-dimensional periodic pinning structure 共shown by vertical lines兲; up
and down arrows denote domains which produce the effective magnetic
field H⫾4 ␲ M 0 in superconducting layers.
neighboring domains inducing vortices and antivortices, see
inset in Fig. 1. In the absence of the external magnetic field
all vortices and antivortices are spontaneous ones; an external magnetic field suppresses antivortices and adds extra vortices. At H⭓4 ␲ M 0 antivortices are absent. In the following
we will discuss this case assuming that 4 ␲ M 0 ⭐H⬍H coer .
The critical current J c is maximal at matching fields when
vortices are positioned in minimums of the pinning potential,
that is, some of the reciprocal vectors of the triangular vortex
lattice coincide with ␲ K/l where K is an integer. These
matching fields are given by the expression
H n 1 n 2 ⫽K 2 共 )/2兲共 ␾ 0 /4l 2 兲共 n 21 ⫹n 22 ⫹n 1 n 2 兲 ⫺1 ,
共2兲
where n 1 and n 2 are integers. At these fields the period of the
vortex lattice is commensurate with the period of the pinning
potential, see Fig. 2. For K⫽1 the maximal matching field is
H m ⬃ ␾ 0 /l 2 , which is of the order 10 G at l⫽1 ␮ m. The
critical current at this field may be found by equating the
pinning force U mp /l to the Lorentz force acting on the vortex line from the transport current in the SC layer, J ␾ 0 d s /c.
This gives
J c ⬃cM 0 /l
共3兲
for the critical current. Not very close to T c 共the condition
for ␭ L Ⰶl to be fulfilled兲 at M 0 ⫽50 emu we obtain J c
⬃107 A/cm2. Obviously, this estimate is correct only if the
magnetic field H m in the sample does not exceed the coercive field H coer .
At H⬃H m (4 ␲ M 0 ⬍H m ⬍H coer), the critical current depends on the relation between the strength of the pinning
potential and the strength of the repulsive vortex–vortex interaction. Vortex–vortex interaction tends to place vortices
in a periodic lattice, which is incompatible with positions
most favorable for pinning. As a result of this competition,
the critical current decreases as vortices become closer to
each other 共K increases兲. For weak pinning 共that is, weaker
than the repulsion between vortices兲 vortices adjust weakly
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2596
Appl. Phys. Lett., Vol. 76, No. 18, 1 May 2000
FIG. 3. Irregular domain structure in a TbFe2 ferromagnetic film of thickness 1.5 ␮m after rotating it in a slowly decaying magnetic field 共Ref. 7兲.
to the pinning potential and the decrease of J c is exponential
in K, see Ref. 10. In the case of strong pinning, quantitative
results were obtained for a one-dimensional vortex system
only.9 The main parameter in this case is the ratio of the
elastic modulus to the strength of the pinning potential, ␤
⫽C/q 2 U mp , where q is the wave vector of the periodic potential, q⫽ ␲ /l in our case. The elastic 共compression兲 modulus C in a one-dimensional structure is determined by the
vortex repulsion. The pinning is weak or strong depending
on whether ␤ Ⰷ1 or ␤ Ⰶ1. At 2 ␲ ␤ ⭐1 pinning is strong and
the critical current remains almost the same as at H m . It
drops by a factor of 100 for ␤ ⬇1 when the magnetic field is
not close to H m . For the two-dimensional vortex system the
critical current in the case of strong pinning has not yet been
calculated. In that case the stiffness of the vortex lattice is
determined by the compression modulus C 11⬇B 2 /4␲ and by
the shear modulus C 66⫽B ␾ 0 /(8 ␲ ␭ L ) 2 , the shear modulus
being smaller for high fields. The adjustment of the vortex
lattice to the pinning potential depends on both moduli. A
rough estimate for the critical current may be obtained by
taking the effective elastic modulus, C⫽ 关 C 66C 11兴 1/2 and using the above mentioned results of the one-dimensional
model.9 Then at B⫽0.1 T, l⫽1 ␮ m, M 0 ⫽50 G, and ␭ L
⫽0.15 ␮ m we obtain 2 ␲ ␤ ⬇1 and the critical current remains almost the same as at H m . We see that it is the coercive field H coer that mostly limits the critical current of
FM/SC multilayers in the presence of a magnetic field.
In real situations, domains in FM layers do not exhibit
perfect periodic stripe structure but are curved due to thermal
Bulaevskii, Chudnovsky, and Maley
fluctuations11 and due to pinning of domain walls by crystal
defects, see Fig. 3. In a multilayered system, the domain
structure will vary from one layer to another. On one hand,
this disorder gives pinning for any orientation of the supercurrent and it eliminates the commensurability effect. On the
other hand it leads to a decrease of the critical current. For
strong pinning this decrease should not be drastic, however.
We note that when SC layers have thickness greater than
␭ L and the lower critical field H c1 ⬍4 ␲ M 0 , spontaneous
vortices connecting domains with opposite magnetization are
induced, leading to an interesting novel vortex state. Note
also that in magnetic fields above coercive field domain
structure disappears and magnetic pinning vanishes abruptly.
Thus FN/SC multilayers may have applications as switching
devices.
In conclusion, we suggest that SC/FM multilayers of
nanoscale period can provide strong pinning of vortices due
to their magnetic interaction with the domain structure in FM
layers in the magnetic fields below the coercive field.
This work has been supported by the DOE Grant No.
DE-FG02-93ER45487. The work in Los Alamos was supported by DOE.
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