Competing symmetries and broken bonds in superconducting vortex-antivortex molecular crystals.
ABSTRACT Hall probe microscopy has been used to image vortex-antivortex molecules induced in superconducting Pb films by the stray fields from square arrays of magnetic dots. We have directly observed spontaneous vortex-antivortex pairs and studied how they interact with added free (anti)fluxons in an applied magnetic field. We observe a variety of phenomena arising from competing symmetries which either drive added antivortices to join antivortex shells around dots or stabilize the translationally symmetric antivortex lattice between the dots. Added vortices annihilate antivortex shells, leading first to a stable "nulling state" with no free fluxons and then, at high densities, to vortex shells around the dots stabilized by the asymmetric antipinning potential. Our experimental findings are in good agreement with Ginzburg-Landau calculations.
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ABSTRACT: The microscopic mechanism of the matching effect in a superconductor, which manifested itself as the production of peaks or cusps in the critical current at specific values of the applied magnetic field, was investigated with Lorentz microscopy to allow direct observation of the behavior of vortices in a niobium thin film having a regular array of artificial defects. Vortices were observed to form regular and consequently rigid lattices at the matching magnetic field, at its multiples, and at its fractions. The dynamic observation furthermore revealed that vortices were most difficult to move at the matching field, whereas excess vortices moved easily.Science 12/1996; 274(5290):1167-70. · 31.20 Impact Factor
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ABSTRACT: Colloidal suspensions are widely used to study processes such as melting, freezing and glass transitions. This is because they display the same phase behaviour as atoms or molecules, with the nano- to micrometre size of the colloidal particles making it possible to observe them directly in real space. Another attractive feature is that different types of colloidal interactions, such as long-range repulsive, short-range attractive, hard-sphere-like and dipolar, can be realized and give rise to equilibrium phases. However, spherically symmetric, long-range attractions (that is, ionic interactions) have so far always resulted in irreversible colloidal aggregation. Here we show that the electrostatic interaction between oppositely charged particles can be tuned such that large ionic colloidal crystals form readily, with our theory and simulations confirming the stability of these structures. We find that in contrast to atomic systems, the stoichiometry of our colloidal crystals is not dictated by charge neutrality; this allows us to obtain a remarkable diversity of new binary structures. An external electric field melts the crystals, confirming that the constituent particles are indeed oppositely charged. Colloidal model systems can thus be used to study the phase behaviour of ionic species. We also expect that our approach to controlling opposite-charge interactions will facilitate the production of binary crystals of micrometre-sized particles, which could find use as advanced materials for photonic applications.Nature 10/2005; 437(7056):235-40. · 38.60 Impact Factor
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ABSTRACT: Novel vortex structures are found when a thin superconducting (SC) film is covered with a lattice of out-of-plane magnetized magnetic dots (MDs). The stray magnetic field of the dots confines the vortices to the MD regions, surrounded by antivortices which "crystallize" into regular lattices. First- and second-order transitions are found as the magnetic array is made sparser or MD magnetization larger. For sparse MD arrays fractional vortex-antivortex states are formed, where the crystal symmetry is combined with a nonuniform "charge" distribution. We demonstrate that due to the (anti)vortices and the supercurrents induced by the MDs, the critical current of the sample actually increases if exposed to a homogeneous external magnetic field, contrary to conventional SC behavior.Physical Review Letters 01/2005; 93(26 Pt 1):267006. · 7.94 Impact Factor
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Neal, J.S., Milosevic, M.V., Bending, S.J., Potenza, A., Emeterio, L.S. and
Marrows, C.H. (2007), Competing symmetries and broken bonds in
superconducting vortex-antivortex molecular crystals, Physical Review Letters,
Volume 99 (127001).
Competing symmetries and broken bonds in superconducting
vortex-antivortex “molecular crystals”
J. S. Neal, M. V. Miloˇ sevi´ c,∗and S. J. Bending†
Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK
A. Potenza,‡L. San Emeterio, and C. H. Marrows
School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK
(Dated: July 17, 2007)
Hall probe microscopy has been used to image vortex-antivortex “molecules” induced in supercon-
ducting Pb films by the stray fields from square arrays of magnetic dots. We have directly observed
spontaneous vortex-antivortex pairs and studied how they interact with added “free” (anti)fluxons
in an applied magnetic field. We observe a variety of phenomena arising from competing sym-
metries which either drive added antivortices to join antivortex shells around dots or stabilize the
translationally symmetric antivortex lattice between the dots. Added vortices annihilate antivortex
shells, leading first to a stable ‘nulling state’ with no free fluxons and then, at high densities, to
vortex shells around the dots stabilized by the asymmetric anti-pinning potential. Our experimental
findings are in good agreement with Ginzburg-Landau calculations.
PACS numbers: 74.78.Na, 74.25.Ha, 74.20.De.
Two-dimensional ordering and crystallization of parti-
cles on structured substrates has attracted considerable
attention in the past decade. For example, the rich crys-
talline states of colloidal particles have been examined
both theoretically and experimentally , as well as the
ordering of atoms on corrugated surfaces  and vortices
in superconductors with periodic pinning arrays . In
such cases particles are grouped at substrate potential
minima, and each of these groups can act as a single par-
ticle with internal degrees of rotational freedom, forming
states that have additional long-range translational or-
der. The existence of these competing symmetries gives
rise to particularly subtle phenomena and leads to or-
dered states which are analogous to “molecular crystals”.
The question of crystallization becomes particularly in-
teresting when single species molecules are replaced by
ionic ones containing positive and negative counterparts.
Recently, colloidal crystals of oppositely charged parti-
cles have been experimentally realized . Surprisingly,
it was found that the stoichiometry of such crystals is
not dictated by charge neutrality, allowing the forma-
tion of a diverse range of binary structures, which grad-
ually melted upon application of an electric field. Analo-
gous “ionic” structures can be found in superconductors;
specifically in superconducting films deposited on spa-
tial arrays of magnets. Each magnet may generate one
or more spontaneous vortex-antivortex (V-AV) pairs in
the superconducting film. These either remain associ-
ated with individual magnets as V-AV “molecules” 
in dilute arrays, or organize themselves into an “ionic”
crystal in dense arrays . To date there has been no ex-
perimental verification of such spontaneous V-AV struc-
tures, which is the first objective of this Letter. Exactly
how V-AV molecules transform into lattices (analogous
to ionic colloidal crystals), and how they interact with
(anti)fluxons introduced by external magnetic fields re-
main challenging questions for both theory and experi-
ment, and this work yields critical insights in these areas.
In this Letter, we directly study V-AV structures
in a Pb superconducting film deposited on a square
array of magnetic dots with perpendicular magnetiza-
tion (see Fig. 1), in an applied homogeneous magnetic
field. Superconductor-ferromagnet hybrid systems can
be broadly divided into two classes - those where mag-
netic nanostructures with weak moments are used as pin-
ning sites to enhance the superconducting critical current
by suppressing flux line motion , and those with strong
moments which lead to the spontaneous formation of V-
AV pairs. The latter have been found to enhance the
critical temperature of the film at finite magnetic field
through field compensation effects . While such consid-
erations are valid near the superconductor-normal phase
boundary, where screening can be neglected (jc≈ 0), the
situation deep within the superconducting state is qual-
itatively different owing to the requirement for magnetic
flux to be quantized. Here a simple picture of field can-
celation is no longer applicable and a microscopic picture
FIG. 1: AFM image of the magnetic disk array (left) and (i)-
(vi) SHPM images of magnetization reversal (scan range is ∼
17µm × 17µm at 20K). The grayscale of images (i) and (ii)
spans 2.66G and 2.63G respectively (see text).
of the formation of spontaneous V-AV pairs is essential
along with an understanding of annihilation and trap-
ping processes. This is a regime where, to date, very few
experimental studies have been made (c.f. ).
To address the important outstanding issues in the
low-temperature regime, we have performed high spatial
resolution scanning Hall probe microscopy (SHPM) 
on hybrid samples deep inside the superconducting state
(0.67 < T/Tc< 1). The samples investigated consisted
of a 1.5mm × 1.5mm array of ferromagnetic disks covered
with a type II superconducting Pb film. The disks were
formed in a [Co(0.5nm)/Pt(1nm)]12multilayer film sput-
tered on a Si/SO2substrate with uniaxial perpendicular
magnetic anisotropy. They were patterned by electron
beam lithography and reactive ion etching through an
evaporated Al etch mask. Four different diameter cir-
cular disks with different magnetic moments have been
patterned on the corners of a 5µm × 5µm square cell
which was repeated periodically in a square lattice, al-
lowing the behavior of dots with different spontaneous
V-AV numbers to be compared in the same sample. De-
sign diameters of 522nm (dot A), 738nm (D), 808nm (B)
and 902nm (C) were chosen, corresponding theoretically
to 1, 3, 3 and 5 spontaneous V-AVs respectively . Fig.
1 shows an atomic force micrograph of the unit-cell of the
disk array. The unpatterned Co/Pt film was measured
by the Magneto-Optical Kerr Effect (MOKE) at 300K
and shown to have high remanence and a coercive field
of ∼ 1000Oe. Fig. 1 shows SHPM images of magnetiza-
tion reversal in the disks at T = 20K, indicating a range
of coercive fields spanning 700-1000Oe and magnetic sat-
uration above ∼ ±1000Oe. Switching of the weakly cou-
pled disks is largely uncorrelated, but once magnetized,
disks of a given size exhibit very strong remanence at
H = 0 and remain in a single domain state with highly
uniform out-of-plane moments. The disks were coated
with a 20nm Ge layer to suppress proximity effects and
an 80nm Pb film deposited using dc magnetron sput-
tering followed by a 10nm Mo capping layer to prevent
oxidation. Magnetization measurements on a single Pb
film of the same thickness indicate that it is a type II
superconductor with Tc = 6.68K, λeff(5K) ≈ 120nm
and ξ(5K) ≈ 50nm. Finally the sample was also coated
with 20nm Ge and 50nm Au to enhance the stability of
the SHPM when in tunneling contact. Microscopy was
performed in a 7T superconducting magnet at T = 5K
with a ∼ 0.5µm spatial resolution GaAs/AlGaAs Hall
sensor. Prior to imaging the Co/Pt dots were magne-
tized to saturation in an applied magnetic field of 3000Oe.
An unwanted consequence of this was a small amount of
trapped magnetic flux in our superconducting solenoid,
with a remanent field ≈ −3.5Oe acting in the opposite
direction to the (positive) dot magnetization. This ‘back-
ground field’ is estimated from a comparison between im-
ages and simulations (c.f. spontaneous V-AVs in Fig. 2
and “nulling” state in Fig. 4), and creates a constant off-
0 1 2 3 4 5
0 1 2 3 4 5
FIG. 2: (Color online) SHPM image of spontaneous V-AV
configurations at Ha = 3.5Oe (Heff ≈ 0) and T=5K (a), and
a schematic depiction of the AV locations (b) (dashed circles
indicate the locations of magnetic disks). (c) Map of magnetic
induction across theoretically predicted V-AV configurations.
(d,e) Magnetic induction map across dot B, experiment (d)
vs. theory (e). (f,g) Induction profiles across one AV along
the lines indicated in (d).
set to our applied field (Ha) axis in all cases. Defining
the actual applied field as Heff(∼= Ha−3.5Oe) there are
two noteworthy field conditions - the spontaneous V-AV
state when Heff= 0, and a “nulling” state (Hnull) when
all the spontaneous AVs have been exactly annihilated
by externally added flux quanta. For Heff < 0 we have
excess ‘free’ AVs, for 0 < Heff< Hnullwe have a gradual
annihilation of spontaneous AVs, and for Heff > Hnull
we have free vortices.
Zero effective applied field.
taneous V-AV configurations at Heff ≈ 0. Fig. 2(a)
shows the first direct observation of spontaneous V-AV
shell structures. A strongly non-linear grayscale has been
used to enhance identification of AVs. This can lead to
apparent variations in AV intensity due to small varia-
tions in e.g. scan height, but does not influence our anal-
ysis which is based on identification of discrete fluxons.
This SHPM image maps the full scan range of our mi-
croscope (13µm × 13µm) and, in common with all other
images presented here, was obtained after field-cooling.
As expected, the (black) AVs clearly order in shell-like
structures around the magnetic dots, while (white) vor-
tices remain confined above the dots. Careful line-scan
analysis allows one to determine the exact locations of
AVs and these are sketched for clarity in Fig. 2(b). Fig.
2(c) illustrates the results of Ginzburg-Landau (GL) sim-
ulations for our exact sample geometry, obtained with co-
herence length ξ(0) = 50nm and uniform magnetization
of the dots of M = 750G. Three dimensional calcula-
tions have also been performed to investigate the role
of the topography introduced by the underlying disk ar-
ray (for details of the approach we refer to Ref. ).
The experimentally observed vorticity is in good agree-
We focus first on spon-
ment with simulations, and broadly speaking increases
with the magnetic moment of the disks (subject to flux
quantization). Moreover, the agreement between experi-
ment and theory is further apparent in Figs 2(d,e), which
compare magnetic induction maps across magnet B. To
highlight the structure of one of the bound AVs, Figs
2(f,g) show linescans of the induction profile in the two
indicated orthogonal directions.
Negative effective applied fields.
the effect of introducing additional “free” AVs into
the system by applying negative effective applied fields,
< 0. Panel (a) shows the ‘difference’ image ob-
tained after subtracting image 2 at Heff ≈ −1Oe from
image 1 at Heff ≈ 0Oe. White spots in the difference
image represent either unmatched AVs or ‘annihilated’
Vs in image 2. We see that two new AVs occupy inter-
stitial sites between magnets while a third one joins the
AV shell around dot D. The fourth remaining white spot
cannot be associated with an AV, since it is located un-
der the magnet itself. We conclude that, together with
the adjacent black spot it represents a V-AV pair which
has collapsed. In other words, the 738nm magnetic dot
(D) which induced three V-AV pairs at Heff≈ 0Oe, now
generates only two in the sample at Heff ≈ −1Oe. In
Figs. 3(b-e) we present a series of GL simulations to clar-
ify this point. These illustrate Cooper-pair density plots
obtained at Heff= 0, -0.6, -1, and -1.8Oe respectively.
These figures clearly illustrate how the square sym-
metry of the underlying magnetic lattice imposes itself
on the natural shell structure of the individual V-AV
molecules. For example, in Fig. 3(c), with 3 added exter-
nal AVs, one of the spontaneous AVs detaches from mag-
net B and joins the interstitial AV-lattice. Effectively, a
spontaneous V-AV bond is broken, as the AV opts for the
mutual interaction with other AVs rather than with its
positive counterpart. In a reversal of this scenario for a
larger number of external AVs (e.g. five in Fig. 3(d)),
the excess AV, not needed in the interstitial lattice, ap-
proaches dot D, attracted by the positive core, and joins
the AV shell. This does not, however, mean that the shell
AV structure prevails over the square lattice. Quite the
contrary, the new negatively charged V-AV molecule now
acts as as a single component of the lattice. This is best
illustrated in Fig. 3(e) (with nine external AVs), where
all molecules have negative net charge (A:-2, B:-1, C:-1,
Positive effective applied fields.
of the V-AV molecular crystal in a positive applied field
is more intuitive, as it is mainly governed by the annihi-
lation between AV shells and externally added vortices.
With increasing applied field, each of the molecules pro-
gressively loses its negative “ions”, and becomes posi-
tively charged. However, even after all AVs are annihi-
lated, the vortex “charge” of individual magnetic dots
keeps increasing due to the attraction between a magnet
and a vortex when their moments are parallel . This
The general behavior
FIG. 3: (Color online) (a) SHPM images obtained at two
different effective applied fields imaged at T=5K and their
difference image (see text). (b-e) Cooper-pair density plots
obtained theoretically for applied fields Heff = 0, -0.6, -1,
and -1.8Oe, respectively.
is emphasized in Fig. 4, where we show the number of ex-
perimentally measured off-site fluxons as a function of ap-
plied field, as well as the results of GL calculations. Both
plots clearly show a “nulling” field (Ha≈ 6Oe), where we
have no free fluxons. Importantly we find that this condi-
tion is met for a range of applied fields, i.e. (∆Ha≥ 1Oe).
This ‘locking’ behavior, which arises due to the change in
vortex occupation number of the magnetic disks, ensures
the absence of any off-site fluxons, and consequently en-
hances the critical current of the sample. The asymmetry
of our magnetic array cell is actually very beneficial here
as it ensures pinning of all individual vortices added to
the system and prevents their off-site ordering for non-
commensurate numbers (so-called fractional matching).
Upon further increase of applied field, the non-uniform
changes in on-site vortex occupation across the sample
impact on off-site vortices. The potential landscape for
the pinning of ‘free’ interstitial vortices becomes ‘dy-
namic’ since interactions with pinned vortex molecules
relocate the energy minima as their charge changes. All
the above effects are illustrated in the series of differ-
ence images, Figs. 4(i-iv), between the two indicated
successive field-cooled states. Fig. 4(i) shows the anni-
hilation of AVs (white spots) together with a decrease
in the vortex occupation number of dot D (black spot).
Fig. 4(ii) shows the change in occupation of the two lower
FIG. 4: (Color online) Number of experimentally observed
free (anti)vortices vs.applied magnetic field (open dots)
at T=5K. Solid line shows the predictions of GL theory
(shifted by +4Oe on the field axis to simulate flux trapping in
solenoid). (i)-(iv) Experimental SHPM difference images con-
structed between the indicated fields (see text). ∆H = 0.5Oe
corresponds to ∼3φ0 per field-of-view on average. (v) Exper-
imental SHPM image of the ‘nulling’ state (Ha = 5.5Oe).
FIG. 5: (Color online) SHPM images at T=5K and corre-
sponding Cooper-pair density plots illustrating formation of
(a) an interstitial vortex lattice (Heff ≈ 4.5Oe), and (b) inter-
stitial vortex shells for high vortex densities (Heff ≈ 7.5Oe).
disks near Hnull(white spots), Fig. 4(iii) shows the crys-
tallization of interstitial vortices (white spots), and Fig.
4(iv) shows the interaction-driven movement of vortices
(adjacent pairs of black and white spots). Note that the
off-center incorporation of new on-site vortices in Fig.
4(ii) is strongly indicative of a multi-vortex state above
magnetic disks, in agreement with GL calculations.
Further proof that these phenomena arise due to com-
peting interactions and not disorder is given in Fig. 5,
which illustrates interstitial vortex structures at larger
positive magnetic fields. At sufficiently large magnetic
fields a square interstitial vortex lattice is recovered (Fig.
5(a)) mirroring conventional matching phenomena. How-
ever, the occupation number of the vortex “molecules” at
each of the dots is different (A:2, B:5, C:6, D:4) which
slightly distorts the lattice. The influence of multi-quanta
vortices at the magnet sites becomes more evident at still
higher vortex densities. The presence of four different re-
pulsive potentials propagating radially from the corners
of the square cell and strong interactions between inter-
stitial vortices results in their arrangement in shells (c.f.
Fig. 5(b)). Such an unusual ordering of vortices was
never observed in the presence of uniform pinning. While
AVs form shells around confined vortices due to their mu-
tual attraction, uniquely and counter-intuitively, vortex
shells are formed by repulsion. The same scenario applies
generally to any system of interacting particles in a sim-
ilar environment. Moreover, the complex V-AV interac-
tions demonstrated in this Letter should also be reflected
in two-component colloidal suspensions with oppositely
charged particles, e.g. coated by charged polymers .
In conclusion, we have directly imaged spontaneous V-
AV shell structures induced in superconducting films by
the stray fields of magnetic arrays for the first time. We
observe a variety of subtle phenomena which arise from
competition between the n-fold rotational symmetry of
the V-AV “molecules” and the translationally symmetric
lattice of magnets. Our measurements agree with G-L
calculations and give unique insights into the properties
of ionic crystals based on the ordering of binary systems
of particles, e.g. oppositely charged colloidal particles.
This work was supported by UK-EPSRC grant No.
GR/D034264/1 and PhD studentship GR/P02707/01.
M.V.M. is a Marie-Curie fellow at University of Bath.
∗Also at: Departement Fysica, Universiteit Antwerpen,
Groenenborgerlaan 171, B-2020 Antwerpen, Belgium
†Electronic address: firstname.lastname@example.org
‡Current address: Diamond Light Source Ltd, Diamond
House, Chilton, Didcot, Oxfordshire, OX11 0DE, UK
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