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Mott insulator phases and first-order melting in Bi2Sr2CaCu2O8+
␦
crystals
with periodic surface holes
S. Goldberg,1Y. Segev,1Y. Myasoedov,1I. Gutman,1N. Avraham,1M. Rappaport,1E. Zeldov,1T. Tamegai,2C. W. Hicks,3
and K. A. Moler3
1Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
2Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
3Department of Physics and Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA
共Received 6 August 2008; revised manuscript received 23 January 2009; published 24 February 2009兲
We measured the effects of periodic surface holes, created using a focused ion beam, on the phase diagram
of the vortex matter in high-TcBi2Sr2CaCu2O8+
␦
crystals. Differential magneto-optical measurements show
that the irreversibility line is shifted to higher fields and temperatures with respect to the pristine melting line.
The irreversibility line displays weak field dependence between integer matching fields indicating multiple-
flux-quanta pinning at holes. We find reduced equilibrium compressibility of the vortex matter at integer
matching fields, which is strong evidence for the existence of thermodynamic Mott insulator phases. Shaking
with a transverse ac field surprisingly reveals first-order melting that is not shifted with respect to the pristine
melting line and that seems to occur within the Mott insulator regions. This melting is understood to be the
first-order transition in the bulk of the crystal beneath the surface holes. The transition is visible at the surface,
despite the reduced vortex compressibility in the top layer.
DOI: 10.1103/PhysRevB.79.064523 PACS number共s兲: 74.72.Hs, 74.25.Ha, 74.25.Bt
I. INTRODUCTION
Experimental and theoretical studies of high-Tcmaterials
with correlated pinning centers have led to the discovery of
many novel phases of vortex matter, nonexistent in the pris-
tine materials. These new phases arise from the complex
interplay among intrinsic point disorder, correlated disorder,
vortex-vortex interaction, and temperature. Correlated disor-
der in high-Tccrystals is often introduced by heavy-ion irra-
diation along the crystallographic caxis, which leads to a
random distribution in the a-bplane.1,2The effects of such
random correlated disorder on the high-Tcphase diagram are
relatively well understood, both theoretically3–8and
experimentally.9–12 Measurements of crystals with periodic
correlated disorder are limited since it is not yet known how
to physically realize periodic correlated disorder in thick
samples. Experimental studies of periodic disorder must
therefore choose between two options: study of thin samples
or study of artificial pins located at the sample surface. Ef-
forts have focused mainly on the study of thin superconduct-
ing films with periodic pinning centers.13–23 However, thin
films do not necessarily retain the thermodynamic properties
of the bulk crystal due to enhanced point disorder. Compari-
son of the resulting vortex phases to the thermodynamic
phases of the pristine material is thus usually not possible.
Theoretically, the thermodynamics of vortices in the pres-
ence of random correlated disorder have been studied exten-
sively. Nelson and Vinokur3mapped the pinned vortex mat-
ter onto a system of quantum two-dimensional 共2D兲bosons.
They predicted the Bose glass transition from the low-
temperature Bose glass phase, in which vortices are local-
ized, to a higher-temperature delocalized vortex phase. They
also discussed the possibility of an incompressible Mott in-
sulator 共MI兲phase when the magnetic induction of the
sample Bexactly equals the matching field B
=
0, where
is the density of the columnar defects 共CDs兲. Radzihovsky5
extended this model to include additional phases for B⬎B
:
a low-temperature weak Bose glass phase, in which both
vortices residing at pins and those at interstitial sites are lo-
calized, an interstitial liquid phase at intermediate tempera-
ture, in which interstitial vortices are free to move but those
at pinning sites are still pinned, and a homogenous liquid
phase at higher temperature, in which all vortices are delo-
calized. For periodic pinning centers, the various Bose glass
phases may be modified.24 For periodic surface holes, even
such a modified description is expected to be valid only
within some finite depth from the surface of the supercon-
ductor.
Simulations of two-dimensional systems containing peri-
odic pinning centers that allow only single-vortex occupancy
demonstrate commensurate states at integer matching fields
nB
, with permitted values of ndepending on the geometry
of the pinning centers.25 Solutions of Ginzburg-Landau
theory reveal additional commensurate states with multi-
quanta vortices26,27 for more general sample and pinning
center parameters. Different melting scenarios have been
demonstrated for triangular and kagome arrays at low match-
ing fields28 and for square pinning arrays both at and in be-
tween matching fields.29 The square pinning array at the first
matching field displays three phases: a low-temperature
pinned solid with square geometry, an unpinned 共“floating”兲
solid with triangular geometry, and a high-temperature liquid
that lacks long-range order. At higher commensurate match-
ing fields, the floating solid phase is not found, but an inter-
mediate phase with mobile interstitial vortices similar to the
liquid is observed. Incommensurate fields display a pinned
phase at low temperatures with extra vortices located at in-
terstitial positions, a phase at intermediate temperatures in
which some vortex motion is present with both interstitials
and pinned vortices participating, and a phase at higher tem-
peratures, in which all vortices are mobile. The temperature
at which mobility is observed for incommensurate fields is
PHYSICAL REVIEW B 79, 064523 共2009兲
1098-0121/2009/79共6兲/064523共10兲©2009 The American Physical Society064523-1
lower than the melting temperature at the commensurate
matching fields.29 For the triangular and kagome geometries,
melting at the first matching field involves a low-temperature
pinned solid and a high-temperature liquid only.
Intermediate-temperature phases, in which some or all of the
interstitial vortices are mobile, are observed at higher match-
ing fields.28 These melting transitions are expected to be
most relevant to high-Tcsuperconductors.29
Direct imaging experiments of low-Tcthin films with
artificial periodic disorder have shown that highly ordered
vortex states exist at integer nB
and fractional 共p/q兲B
matching fields, with n,q, and pintegers.30,31 Due to
these ordered vortex states, such films have demonstrated
commensurate effects in critical current,13–16
magnetization,17 magnetoresistance,18,19 and magnetic-
susceptibility measurements.20–23 Possible phases and phase
transitions of the vortex matter have been inferred from these
measurements. Enhanced flux creep rate for B⬎B
was
thought to be evidence of a transition from an incompressible
MI state to an interstitial liquid state.17 Shapiro steps in
transport measurements were understood to be a result of the
coexistence of vortices pinned to artificial pinning sites and
mobile interstitial vortices.32 The behavior of the critical cur-
rent was interpreted as evidence of two depinning energies,
corresponding to the upper boundaries of the weak Bose
glass and interstitial liquid phases.19 Onsets of nonzero real
and imaginary parts of the magnetic susceptibility were ten-
tatively identified as the lower and upper phase boundaries of
an interstitial liquid phase.22 In addition to these states, the
possibility of a saturation number ns⬎1, corresponding to ns
vortices at each pinning site, leads to multiple-quanta pinned
vortex states, which have been observed in many
samples.15,23
There are fewer experimental data regarding the thermo-
dynamic phases of high-Tcsuperconductors with periodic ar-
tificial pinning centers. The critical current in YBa2Cu3O7
共YBCO兲thin films exhibited integer commensurate effects33
over a large temperature range. Scanning Hall probe mea-
surements indicated that trapping of ⬃15 flux quanta is pos-
sible for 2.5-
m-diameter holes close to Tcin YBCO.34 Thin
crystalline Bi2Sr2CaCu2O8+
␦
共BSCCO兲samples with fully
penetrating periodic holes exhibited integer35 and rational36
matching effects, in magnetoresistance and transport mea-
surements, respectively. Similar samples with surface holes
also displayed matching effects in magnetoresistance and
critical current.37 A single study using thick BSCCO samples
with surface holes displayed integer matching in local
magnetization.38 In these studies of BSCCO, the matching
effects were visible in the field and temperature ranges at
which the vortex matter is known to be in a liquid state in
pristine crystals. No first-order melting step39 was measured
in these samples, and a full description of thermodynamic
phases and transitions is lacking.
In this work, we present an investigation of a thick
BSCCO crystal, partially patterned with periodic surface
holes created by a focused ion beam, measured using differ-
ential magneto-optics 共DMO兲共Refs. 40–42兲accompanied by
shaking with transverse ac field.43 We observe steplike be-
havior of the irreversibility line 共IL兲, which may be a result
of multiquanta pinning to holes. We see a reduction in the
DMO signal at integer matching fields, evidence of MI
phases. We find a first-order melting transition 共FOT兲in the
patterned regions that is not shifted with respect to the pris-
tine melting line. This FOT is observed even at integer
matching fields, where the vortex matter in the surface layer
is essentially incompressible. We believe this FOT to be the
melting transition of the bulk of the crystal beneath the pat-
terned surface.
II. EXPERIMENTAL DETAILS
Several samples were prepared and studied. Here we
present a detailed investigation of a 2750⫻740⫻30
m3
BSCCO crystal 共Tc⯝90.5 K兲, with two triangular arrays of
periodic holes patterned on the top surface using an FEI
Strata 400 focused ion-beam system. Figures 1共a兲and 1共b兲
show SEM images of the hole profile and periodicity, respec-
tively. The measured hole depth was approximately 1.4
m.
Hole diameter decreases from ⬃0.6
m at the sample sur-
face to ⬃0.3
m at a depth of 0.7
m. The lattice constant
of both arrays was 0.9
m, corresponding to a matching
field of B
=29.5 G. The dimensions of each array were ap-
proximately 170⫻170
m2.
DMO measurements were performed by modulating the
applied field H
储
caxis by ⌬H=1 Oe while sweeping tem-
perature Tat constant Hor scanning Hat constant T. Each
measurement point required averaging over kcharge-coupled
device 共CCD兲camera exposures, first at H+⌬H/2 and then
at H−⌬H/2, and calculating a difference image. Each DMO
image is the average of msuch difference images. Using
k,m⬃10 with a typical exposure time of 0.3 s yielded a
typical modulation frequency of ⬃0.33 Hz. Values of
dB/dH were derived from the DMO images by dividing the
1µm
a
1µm
b
c
→
M2
M1
100 µm
FIG. 1. 共a兲Scanning electron microscopy 共SEM兲image of the
cross section of the surface holes. Hole depth is ⬃1.4
m. 共b兲Part
of one of the two arrays patterned on the sample, imaged by SEM.
The distance between holes is 0.9
m. 共c兲A DMO image of the
sample at T=80 K and H=21 Oe. The sample’s edge is the verti-
cal border indicated by the arrow. The two B
=29.5 G arrays, M1
and M2 of 170⫻170
m2, appear as darker regions due to their
enhanced irreversibility. The irregular shape of M1 is a result of
surface damage that occurred after writing the holes.
GOLDBERG et al. PHYSICAL REVIEW B 79, 064523 共2009兲
064523-2
local light intensity by the intensity of some region far from
the sample, where it was assumed that dB/dH=1 G/Oe. For
quantitative data analysis, intensities were spatially averaged
over a typical area of ⬃50⫻50
m2. As described
previously,40–42 the DMO measurement with field modula-
tion is essentially equivalent to the measurement of the real
component of the low-frequency local ac susceptibility as
obtained, e.g., by Hall sensors.44 Figure 1共c兲shows a DMO
image of part of the sample taken at T=80 K and H
=21 Oe. The average brightness of the patterned areas M1
and M2 is lower than that of the neighboring pristine sample.
This is due to an elevated IL in the patterned areas as de-
scribed below.
III. RESULTS
We first inspect the IL of the patterned regions. The IL is
important in the context of possible Bose glass phases be-
cause it is thought to be the dynamic manifestation of the
thermodynamic Bose glass transition.3In DMO measure-
ments, reversibility of the vortex matter is quantified by
modulating the applied field by ⌬Hand measuring dB/dH,
the change in the local magnetic induction due to the modu-
lation. Strong pinning results in dB/dH=0, whereas full re-
versibility corresponds to dB/dH=1 G/Oe. The irreversibil-
ity threshold in the following was chosen arbitrarily at
dB/dH=0.8 G/Oe, with TIL共HIL兲denoting the temperature
共external field兲at which this threshold is reached. We empha-
size that the resulting IL reflects the response of the vortex
system at low frequencies. Transport measurements or DMO
with current modulation could possibly map out additional
boundary lines similar to the delocalization line of vortices
from columnar defects;12 however, such measurements are
beyond the scope of the present study. Figure 2shows
dB/dH for the pristine region 关Fig. 2共a兲兴and patterned re-
gion M1 关Figs. 2共b兲and 2共c兲兴measured by Tscans at con-
stant H.
Focusing on the pristine region 关Fig. 2共a兲兴, we see a series
of sharp peaks in dB/dH with paramagnetic dB/dH⬎1
共light gray, light pink online兲, corresponding to the first-order
melting transition Tmfrom a low-temperature vortex solid to
a high-temperature vortex liquid.40,45 The black dots in Fig.
2共b兲show the pristine melting line Tmextracted from Fig.
2共a兲. The patterned region M1 关Fig. 2共b兲兴, in comparison,
shows no FOT. It does, however, exhibit two notable fea-
tures. First, the IL of region M1 is shifted to higher tempera-
ture. This is seen by focusing on TIL 共white contour兲in Figs.
2共a兲and 2共b兲.TIL of the patterned region M1 is significantly
greater than TIL of the pristine region, which occurs at tem-
peratures lower than the pristine Tm. The second notable fea-
ture in Fig. 2共b兲is a narrow finger near H=33 Oe, approxi-
mately 1 Oe wide, for 82⬍T⬍87 K, in which dB /dH of
M1 is suppressed.
Figure 2共c兲shows dB/dH of patterned region M1 over a
larger range of Hand T. The pristine melting line Tmis
plotted as black dots for comparison. The IL of M1 is clearly
shifted to higher temperatures. Sharp fingers, or narrow re-
gions of Hin which dB/dH of M1 is highly suppressed,
occur at H=33, 64, 95, 126, and 157 Oe 共denoted by ar-
rows兲, consistent with integer multiples B/B
=1,2,3,4,and
5 of the predicted matching field B
=29.5 G. The minima in
dB/dH as a function of Hat matching fields are of both
dynamic and thermodynamic origin. Enhanced pinning at
matching fields suppresses vortex motion, and hence also
dB/dH. As discussed below, these minima also indicate nar-
row ranges of Hwith reduced equilibrium compressibility
since the compressional modulus c11 is proportional to
dH/dB.46 In addition to the fingers observed at integer B/B
,
we see that TIL between matching fields is steplike, with
TIL共H兲weakly dependent on Hbetween matching fields, and
shifts in TIL共H兲occurring at matching fields. For example,
TIL⬇76.5 K for 3⬍H/B
⬍4 and TIL ⬇74 K for 4
⬍H/B
⬍5共white contour兲.
We now focus on the IL of the patterned regions below
and in the vicinity of the first matching field. Figure 3shows
dB/dH measured simultaneously in the pristine region and in
the patterned regions M1 and M2 for BⱗB
.TIL 共dB/dH
=0.8 G/Oe, white contour兲of the pristine region is located
below the pristine melting line Tm. In region M1 关Fig. 3共b兲兴,
TIL is shifted to higher temperatures at all fields when com-
pared to the TIL of the pristine region 关Fig. 3共a兲兴. An addi-
T
(
K
)
H
(
Oe
)
c
M1
dB/dH (G/Oe)
←5
←4
←3
←2
←1
65 70 75 80 85
40
60
80
100
120
140
160
0.2 0.4 0.6 0.8 1.0 1.2
T
(
K
)
H(Oe)
bM1
80 82 84 86 88
28
30
32
34
36
38
40
42
H(Oe)
apristine
80 82 84 86 88
28
30
32
34
36
38
40
42
FIG. 2. 共Color online兲dB/dH, the change in magnetic induction
due to field modulation of ⌬H=1 Oe, for different sample regions,
taken during Tscans. 共a兲A pristine region. The pristine melting line
Tmappears as a series of light gray 共light pink兲paramagnetic peaks
in dB/dH with values above 1 G/Oe. This particular run was carried
out on a sparse grid in Tand Hof 0.4 K and 0.5 Oe 共denoted by the
lines in the lower left corner兲. The apparent discontinuity in Tmis an
artifact resulting from grid spacing larger than the width of the
melting peak. 共b兲dB/dH of region M1 in the vicinity of the first
matching field, B
=29.5 G. dB/dH values are reduced compared
to 共a兲and a narrow dip appears at H=33 Oe. The location of the
pristine melting line Tmis denoted by black points. 共c兲dB/dH in
the patterned region M1 over a wide range of Tand H. Matching
effects 共denoted by arrows兲are visible at H=33, 64, 95, 126, and
157 Oe, consistent with integer multiples of the predicted B
=29.5 G. The pristine melting line Tmis denoted by black points.
The patches in the data are a result of slightly differing setup pa-
rameters for the different experimental runs.
MOTT INSULATOR PHASES AND FIRST-ORDER MELTING…PHYSICAL REVIEW B 79, 064523 共2009兲
064523-3
tional sharp finger in the IL, extending to TIL⯝87 K, occurs
at H=33 Oe or B=B
. The IL of patterned region M2 关Fig.
3共c兲兴is shifted somewhat less, yet it too displays a sharp
shift to higher Tat the matching field B
. The reason for the
difference in the shift of the IL of the two arrays may be a
result of the difference in the arrays’ locations 共M2 is closer
to the sample’s edge兲or due to some difference in the holes
of the two arrays, which are not identical and may have
different pinning properties. Still, for both arrays the IL is
shifted upward, with an additional sharp finger at B
.In
BSCCO crystals irradiated with low concentration of CDs
the sharp finger is absent. Instead, a kink is observed in the
vicinity of B
which is believed to be the result of depinning
of two different vortex populations. Below B
,TIL is the
temperature at which vortices located at CDs depin. Above
B
,TIL is the depinning temperature of interstitial vortices.12
Also, fractional matching features have been observed37 in
BSCCO samples with periodic surface defects. We do not
detect any fractional matching features in the IL of the pat-
terned regions; the reason for this is not clear but is consis-
tent with results shown elsewhere.38
The IL in BSCCO is known to be a dynamic feature of the
phase diagram.4,47,48 However, since it indicates a region of
the phase diagram in which there is a change in the system’s
dynamic response, it may indicate an underlying thermody-
namic transition that occurs at similar values of field and
temperature. In order to detect a possible underlying FOT,
and to determine whether the reduced dB/dH regions are
truly a thermodynamic feature of the phase diagram, we
studied the behavior of the first matching field in the pres-
ence of shaking. The shaking technique43 utilizes an in-plane
ac magnetic field. It is known to suppress hysteretic behavior
in BSCCO, enabling the observation of thermodynamic
properties.48–50
The effect of shaking is shown in Fig. 4for the pristine
and patterned regions M1 and M2 at 77 K. Increasing shak-
ing amplitude Hac
⬜leads to a systematic increase in dB/dH,
which saturates near 1 G/Oe, as expected for a fully pen-
etrable sample. A feature common to all three plots in Fig. 4,
and thus unrelated to the surface holes, is an abrupt change
from zero to negative dB/dH at ⯝8 Oe. Below ⯝8 Oe, the
sample is in the Meissner phase, with B=0. dB /dH ⬍0 im-
mediately above the Meissner phase corresponds to negative
local permeability.51 This effect, which occurs in BSCCO
samples with platelet geometry, is a result of the geometrical
barrier52,53 and the modulation of the vortex dome during the
modulation cycle of the applied field H⫾⌬H/2. This nega-
tive permeability is also visible as a black strip at low fields
in Fig. 3共a兲. It is interesting to note that the negative dB/dH
values are not visible at the highest shaking amplitude Hac
⬜
=121.8 Oe. This indicates that the shaking field enables the
vortices to overcome the geometrical barrier. Consequently,
the local negative permeability changes to high positive local
permeability, as seen in Figs. 4共a兲and 4共c兲. This is the ex-
pected behavior in the absence of geometrical barriers, as
demonstrated for prism-shaped samples.54
There are two notable differences between the pristine
关Fig. 4共a兲兴and patterned 关Figs. 4共b兲and 4共c兲兴regions. The
first difference is the appearance of the matching feature in
the form of a dip in dB/dH near 32 Oe, denoted by arrows in
Fig. 4. Shaking extends the range of temperatures for which
this dip is visible well into the vortex solid region below Tm.
Without shaking, the first matching feature is not visible at
this temperature 共77 K, see Fig. 2兲due to the enhanced pin-
ning in the vortex solid. With increased Hac
⬜, the matching
feature appears first as a step 关Fig. 4共b兲,Hac
⬜=20.3 and 40.6
Oe兴and then as a dip in dB/dH 共Hac
⬜=81.2 Oe兲. In some
cases, as shown in Fig. 4共c兲for high Hac
⬜, a peak appears in
H (Oe)
a
pristine
80 82 84 86 88
5
10
15
20
25
30
35
40
T
(
K
)
b
M1
80 82 84 86 88
c
M2
80 82 84 86 88
dB
/
dH
(G/Oe)
−0.
2
0.0
0.2
0.
4
0.6
0.8
1.0
FIG. 3. 共Color online兲dB/dH for BⱗB
, measured during T
scans. 共a兲dB/dH of a pristine region. The melting transition Tm
appears as a line with dB/dH⬎1G/Oe 共light gray, light pink on-
line兲. The temperature TIL at which the IL is located 共dB/dH
=0.8 G/Oe, white兲is found below Tm.共b兲and 共c兲show dB/dH of
patterned regions M1 and M2, respectively. TIL of M1 and M2 is
shifted to higher temperatures relative to the pristine TIL. A sharp
finger in TIL of both M1 and M2 appears at H=33 Oe, where B
=B
. Negative values of dB/dH 共black兲correspond to negative
permeability due to geometrical barriers.
0.0
0.5
1.0
1.5 apristine
Hac
⊥
121.8 Oe
81.2 Oe
40.6 Oe
20.3 Oe
0.0 Oe
0.0
0.5
1.0
dB/dH [G/O
e
]
b
M1
→
1
010 20 30 40 50
0.0
0.5
1.0
H [Oe]
c
M2
→
1
FIG. 4. 共Color online兲dB/dH vs applied field H共scanned up
and down兲for different values of in-plane shaking amplitude Hac
⬜at
T=77 K for the 共a兲pristine region and patterned regions 共b兲M1
and 共c兲M2. Arrows denote the first matching field. Data are shown
for different values of applied Hac
⬜共from bottom to top兲:0共䉯兲, 20.3
共䉮兲, 40.6 共䉭兲, 81.2 共〫兲, and 121.8 共䊐兲Oe. Shaking frequency was
15 Hz for all measurements. Symbols appear every 37 data points.
GOLDBERG et al. PHYSICAL REVIEW B 79, 064523 共2009兲
064523-4
dB/dH immediately before the dip. The slight downward
shift in Hof the dip for increasing Hac
⬜probably results from
the increased penetration of magnetic induction Bat higher
Hac
⬜, resulting in the same B=B
at slightly lower values of
applied field H. The dip in dB/dH corresponds to a reduction
in the compressibility of the vortex matter at B
.
The second difference between the pristine and patterned
regions in Fig. 4can be seen away from the matching field.
For the pristine region, the values of dB/dH increase gradu-
ally from dB/dH⯝0 at low field to dB/dH=1 G/Oe at suf-
ficiently high applied field H. For the patterned regions, the
behavior of dB/dH is plateaulike, with the matching feature
dividing between neighboring plateaus. This can be seen in
Fig. 4共b兲for Hac
⬜=81.2 and 121.8 Oe and in Fig. 4共c兲for
Hac
⬜=40.6 Oe. These plateaus are consistent with the ob-
served steplike behavior of TIL during Tscans, as shown in
Fig. 2共c兲. The plateaus in dB/dH appear between matching
fields, where TIL is almost independent of H. The step be-
tween the plateaus appears at H=B
, consistent with the
steps in TIL that occur at H=nB
. The value of dB/dH for
each plateau in Figs. 4共b兲and 4共c兲increases with increasing
Hac
⬜. Figures 5共a兲and 5共b兲show the effects of shaking with
different values of Hac
⬜for B/B
=2 and 3 and B/B
=3 and
4, respectively, for patterned region M2. The data for M1 are
similar. Clearly the same matching features that appear for
B/B
=1, namely, a step in dB/dH for low Hac
⬜that develops
into a dip for higher Hac
⬜are visible also for higher matching
fields.
Interestingly, increasing Tand increasing shaking ampli-
tude Hac
⬜have similar effects on dB/dH immediately below
the IL. This can be seen by comparing Figs. 5共a兲and 5共c兲,in
which dB/dH of region M2 in the vicinity of B/B
=2 and 3
is plotted for different values of Hac
⬜and T, respectively. In-
creasing Hac
⬜and increasing T共from bottom to top curves兲
both tend to increase dB/dH and both have the effect of
transforming the matching feature from a step to a dip. How-
ever, the overshoot in dB/dH immediately below the match-
ing feature appears only at nonzero Hac
⬜. Similar behavior is
observed for B/B
=3 and 4 in Figs. 5共b兲and 5共d兲.
We now address the question of first-order melting within
the patterned regions in the presence of shaking. For the
results shown below, we applied 15 Hz Hac
⬜=81.2 Oe shak-
ing field. We find that shaking shifts the IL to lower fields
and temperatures and thus enables the observation of a FOT
in the patterned regions of the sample. Figure 6shows de-
tailed scans of the H-Tregion in which the pristine Tmline
intersects the B/B
=1 matching line. dB /dH is shown for
both patterned regions and for the pristine region, without
and with shaking 共top and bottom panels, respectively兲. For
the pristine region 关Figs. 6共a兲and 6共b兲兴,dB/dH at lower T
and His raised slightly by shaking and the pristine melting
line Tm, which appears as a line with paramagnetic dB/dH
⬎1G/Oe 共light gray, light pink online兲, remains essentially
unchanged. For the patterned region M1, no FOT was visible
without shaking 关Fig. 6共c兲兴. With shaking 关Fig. 6共d兲兴, a FOT
became visible. It appears to be located at the same tempera-
tures and fields as the pristine melting line Tm. Remarkably,
the Tmline is clearly visible even at the bottom of the B
matching dip. For the patterned region M2, shaking was not
needed to uncover the FOT 关Fig. 6共e兲兴. While shaking raised
dB/dH values overall, it did not change the location or the
nature of the FOT 关Fig. 6共f兲兴. Similar results are shown for
B/B
=2 in Fig. 7. In this case, shaking was necessary to
view the FOT in both patterned regions M1 关Figs. 7共c兲and
7共d兲兴and M2 关Figs. 7共e兲and 7共f兲兴. Note that the location of
the FOT of the patterned regions in the phase diagram is
indistinguishable from the location of the pristine melting
line Tm. Moreover, Tmand the nB
lines seem to intersect
with no apparent interaction, as if the periodic pinning po-
tential of the holes has no effect on melting. No additional
FOT was detected for either of the patterned regions. We
emphasize that at the points in the phase diagram where the
FOT meets the matching fields, a contradictory behavior of
60 70 80 90 100
0.0
0.2
0.4
0.6
0.8
1.0
dB/dH (G/Oe)
a
T=74 K
Hac
⊥
→
2
→
3
81.2 Oe
40.6 Oe
20.3 Oe
0.0 Oe
90 100 110 120 130 140
0.0
0.2
0.4
0.6
0.8
1.0
b
T=70 K
Hac
⊥
→
3
→
4
81.2 Oe
40.6 Oe
20.3 Oe
0.0 Oe
60 70 80 90 100
0.0
0.2
0.4
0.6
0.8
1.0
dB/dH (G/Oe)
H (Oe)
c
Hac
⊥=0Oe
T
→
2
→
3
78 K
76 K
74 K
90 100 110 120 130 140
0.0
0.2
0.4
0.6
0.8
1.0
H (Oe)
d
Hac
⊥=0Oe
T
→
3
→
4
70 K
68 K
76 K
74 K
72 K
FIG. 5. 共Color online兲dB /dH of region M2 as a function of H
for different temperatures Tand shaking amplitudes Hac
⬜. The effect
of shaking is similar to the effect of temperature 共see text兲.共a兲
Matching effects at T=74 K and B/B
=2 and 3 and 共b兲at T
=70 K and B/B
=3 and 4. Hac
⬜=0 共䉮兲, 20.3 共䉭兲, 40.6 共〫兲, and
81.2 共䊐兲Oe. 共c兲Matching effects at T=74 共䉭兲,76共〫兲, and 78 共䊐兲
K; B/B
=2 and 3, without shaking. 共d兲Matching effects at T=68
共䉯兲,70共䉮兲,72共䉭兲,74共〫兲, and 76 共䊐兲K; B/B
=3 and 4,
without shaking. Arrows denote matching fields. Symbols appear
every 30 data points.
H
(O
e
)
a
pristine
81 83 85 87 89
29
30
31
32
33
34
35 c
M1
81 83 85 87 89
e
M2
81 83 85 87 89
29
30
31
32
33
34
35
H
(O
e
)
b
pristine
81 83 85 87 89
29
30
31
32
33
34
35
T
(
K
)
d
M1
81 83 85 87 89
f
M2
81 83 85 87 89
29
30
31
32
33
34
35
dB
/
dH
(G/Oe)
0.2
0.
3
0.
4
0.5
0.
6
0.
7
0.8
0.9
1.
0
1.1
FIG. 6. 共Color online兲dB /dH as a function of Hand Tnear the
intersection of melting and B/B
=1 matching, without 共top panels兲
and with 共bottom panels兲shaking. Shaking parameters were 15 Hz
and 81.2 Oe. Results are shown for the 关共a兲and 共b兲兴 pristine sample
and patterned regions M1 and M2 关共c兲and 共d兲and 共e兲and 共f兲,
respectively兴.
MOTT INSULATOR PHASES AND FIRST-ORDER MELTING…PHYSICAL REVIEW B 79, 064523 共2009兲
064523-5
the vortex lattice occurs. On one hand, at the FOT there is a
jump in vortex density. On the other hand, at matching fields
the vortex matter exhibits a strongly enhanced compressibil-
ity modulus c11 ⬃共dB/dH兲−1 that exists both below and
above Tm. This apparent contradiction is discussed below.
IV. DISCUSSION
In order to understand the observed behavior of the IL, we
consider two possible physical scenarios.55 In the first sce-
nario, shown schematically in Fig. 8共a兲, we assume that each
hole can pin only a single vortex. As a result, two vortex
populations are present for B⬎B
: vortices located at holes
and interstitial vortices located between holes. The interstitial
vortices are subjected to a caging potential caused by the
vortices located at holes24 that is assumed to be weaker than
the pinning potential at holes but stronger than the pristine
pinning. This gives rise to a depinning transition of the in-
terstitials, which we identify with TIL, at a temperature above
the pristine melting temperature Tm. Alternatively, we con-
sider a scenario in which there is a multiquanta pinning by
holes. We assume that below the IL, all vortices are located
at holes, while above the IL, some vortices are depinned
from holes, and thus mobile, as shown schematically in Fig.
8共b兲. Due to repulsion between pinned vortices, the pinning
force per vortex is expected to decrease as a function of the
number of vortices pinned to the hole.56 We therefore assume
that the pinned vortices residing at holes depin one at a time,
as Tis increased. Within this multiquanta scenario, TIL cor-
responds to the temperature at which the first vortices depin
from holes. Either of the two scenarios must provide an ex-
planation for the observed behavior of the IL: the plateaus in
the IL away from matching, the shift in the IL to lower Tand
Hin the presence of shaking, and the sharp dips at the
matching fields.
We begin with the plateaus and steps in the IL. In the
multiquanta scenario, the maximum number of vortices per
hole nmax共T兲is determined by temperature-dependent hole
pinning strength and repulsive interactions between vortices
located at the hole and decreases with increasing
temperature.56 We denote the temperature at which nmax de-
creases from nto n−1 by Tn关see dotted lines in Fig. 8共c兲兴.
We assume that for the applied fields Hⱕ6B
, all vortices
are pinned to holes at sufficiently low T.AsTis increased
above Tn, holes may only pin n−1 vortices. Therefore vorti-
ces abruptly depin from holes occupied by nvortices, leav-
H
(O
e
)
a
pristine
75 78 81 84 87
60
61
62
63
64
65
66
67 c
M1
75 78 81 84 87
e
M2
75 78 81 84 87
60
61
62
63
64
65
66
67
H
(O
e
)
b
pristine
75 78 81 84 87
60
61
62
63
64
65
66
T
(
K
)
d
M1
75 78 81 84 87
f
M2
75 78 81 84 87
60
61
62
63
64
65
66
dB
/
dH
(G/Oe)
0.2
0.
3
0.
4
0.5
0.
6
0.
7
0.8
0.9
1.
0
1.1
FIG. 7. 共Color online兲dB /dH as a function of Hand Tnear the
intersection of melting and B/B
=2 matching, without 共top panels兲
and with 共bottom panels兲shaking. Shaking parameters were 15 Hz
and 81.2 Oe. Results are shown for the 关共a兲and 共b兲兴 pristine sample
and patterned regions M1 and M2 关共c兲and 共d兲and 共e兲and 共f兲,
respectively兴. The slight curving of the matching effect to lower H
for higher Tin 共c兲–共f兲is due to the increased penetration B共H兲at
higher T.
a b
B/Bφ
c
T1
T2
T3
T4
0
1
2
3
B/Bφ
MQS
HL
ILq
d
T1
T2
T
3
T4
0
1
2
3
FIG. 8. 共Color online兲共a兲and 共b兲A schematic view of the vor-
tex matter for B/B
=3, within 共a兲the single-vortex pinning sce-
nario and 共b兲the multiquanta scenario. Gray circles indicate holes.
Black dots indicate vortices. 共a兲In each unit cell of the pinning
lattice, a single vortex is pinned to the hole and two vortices are
interstitial. Below 共above兲the IL interstitials are pinned 共mobile兲.
共b兲In each unit cell, two vortices remain pinned to the hole. Below
the IL the third vortex is also pinned to the hole 共not shown兲. Above
the IL, it is depinned and mobile. 共c兲and 共d兲A schematic descrip-
tion of the steps in the IL, within the multiquanta scenario, in the 共c兲
absence and 共d兲presence of thermal fluctuations or shaking. 共c兲The
maximum number of quanta per hole nmax共T兲共solid black line兲is
expected to decrease as a function of T, resulting in temperatures Tn
共dashed lines兲above which holes pin n−1 vortices only, and addi-
tional vortices become mobile interstitials. Thus for each interval
共n−1兲⬍B/B
⬍n,TIL=Tn.共d兲Thermal fluctuations shift TIL 共solid
line兲from Tn共dashed lines兲to T
˜
n⬍Tn.AtB/B
=n, the vortex
lattice remains stable to the temperature Tndue to a thermodynamic
MI phase. Resulting phases include a multiquanta solid 共MQS兲be-
low TIL, an interstitial liquid 共ILq, white兲above the IL that termi-
nates at T
˜
1, a homogeneous liquid 共HL兲above T
˜
1, and MI fingers at
B/B
=n共black兲.
GOLDBERG et al. PHYSICAL REVIEW B 79, 064523 共2009兲
064523-6
ing n−1 vortices per hole. The depinned vortices are mobile,
leading to a fast onset of reversibility. We therefore identify
TIL=Tnfor 共n−1兲⬍B/B
⬍n. The resulting IL thus displays
steps and plateaus, as shown schematically in Fig. 8共c兲. Fi-
nite temperature and slight hole variability are likely to cause
some variation in nmax共T兲at different holes. This would lead
to some smearing of the irreversibility transition and to a
weak dependence of TIL on Bbetween matching fields due to
different mixing of the Tnas field is varied. This schematic
description neglects the effects of thermal fluctuations, which
will be discussed later on. The IL shown in Fig. 2共c兲共white
contour兲indicates that TIL displays three discrete steps in the
temperature range T=72–77 K. The weak temperature de-
pendence of the IL between matching fields, as well as the
plateaus in dB/dH in Figs. 4and 5, seems to indicate that the
IL is not strongly affected by interactions between vortices at
neighboring holes. Rather, it is governed by the hole pinning
energy and the repulsion between vortices within a single
hole, leading to the steplike TIL. Within the single-vortex
pinning scenario, in contrast, TIL共H兲is the depinning tem-
perature of interstitial vortices that is expected to decrease
rather smoothly with field as the density of interstitials in-
creases.
We now address the shift of the IL to lower temperatures
in the presence of shaking. Shaking and increased thermal
fluctuations seem to have a similar effect on the IL 共see Fig.
5兲. Within the multiquanta scenario both provide a mecha-
nism for hopping of vortices between vacancies at holes,
thus increasing the dynamic dB/dH and decreasing TIL. Be-
tween matching fields and slightly below TIL, the holes are
below their full pinning capacity, so shaking may provide a
way for vortices to hop between holes more easily or to
depin from holes and move to interstitial positions. The pla-
teaulike behavior indicates that this shaking-induced hopping
is roughly independent of the number of vacancies that ex-
ists. Instead, the degree of hopping or depinning is dependent
on the balance between the roughly field-independent pin-
ning energy of the nmaxth vortex and the activation energy of
the applied shaking. Within the single-vortex pinning sce-
nario, above the IL the interstitials are mobile; therefore,
immediately below the IL they are weakly pinned. Shaking
will thus assist in overcoming the weak pinning potential,
decreasing TIL. The existence of the plateaus, however, can-
not be easily explained in this case.
We now address the third experimental finding, namely,
the fingerlike dips in dB/dH at matching fields. Focusing on
Figs. 6and 7, we see that, unlike the rest of the IL, the
temperatures at which the fingers terminate are not affected
by in-plane shaking. This strongly suggests that unlike the
IL, which is a dynamic feature of the phase diagram, the
fingers at matching fields are a thermodynamic feature. A
thermodynamic minimum with dB/dH=0 would indicate a
plateau in the equilibrium B共H兲and a diverging bulk modu-
lus c11⬃
H/
B,46 which are clear signatures of the incom-
pressible MI phase. The finite positive minima observed in
dB/dH 关see Figs. 4共b兲,4共c兲, and 5兴correspond to a reduction
in the positive slope of B共H兲at nB
. These finite values at
nB
may be a consequence of the finite size of the patterned
region, which prevents infinite divergence of c11,orofthe
broadening of dB/dH due to the modulation of ⌬H=1 Oe,
which is at least as wide as the width of the dip. For samples
with random defects, it has been argued that the MI phase is
destroyed by repulsive vortex interactions, possibly retaining
“lock-in” effects such as a finite peak in the bulk modulus
c11.46,57 In our measurements, however, the pinning is or-
dered. At matching fields, pinning energy and vortex-vortex
interactions both stabilize the vortex lattice. In this case, ob-
servation of a MI phase is possible.46,57 We believe that the
sharp matching features we observed in dB/dH at nB
that
are not affected by shaking are a strong indication of ther-
modynamic MI phases.
The fingers of reduced dB/dH, or MI phases, may be
understood in the context of the multiquanta scenario. For
共n−1兲⬍B/B
⬍nand T⬍Tn, in the absence of thermal fluc-
tuations, all vortices are pinned to holes 关see Fig. 8共c兲兴. The
maximum number of vortices that can be pinned nB
/
0is
greater than the actual number of vortices B/
0, resulting in
below-full occupancy, or “vacancies,” at some of the multi-
quanta holes. Thermal fluctuations 关which were neglected in
the schematic description in Fig. 8共c兲兴are expected to lead to
vortex hopping between the vacancies, resulting in enhanced
vortex mobility. Thus, for 共n−1兲⬍B/B
⬍n, vortex dynam-
ics lead to a reduction in TIL from Tn关Fig. 8共d兲, dashed lines兴
to some lower temperature T
˜
n. Exactly at matching, an addi-
tional thermodynamic consideration enters. The total hole
capacity equals the number of vortices, and therefore there is
a finite energy cost for adding an extra interstitial vortex. As
a result, the equilibrium B共H兲will exhibit a plateau over a
finite range of Hthat is not affected by shaking. The corre-
sponding minima in dB/dH, or equilibrium MI phases, may
thus be observed up to Tn关black fingers in Fig. 8共d兲兴.
The fingers of reduced dB/dH may also be partially un-
derstood within the single-vortex pinning scenario. At
matching fields, we expect both the vortices at holes and the
interstitials to be ordered in a configuration commensurate
with the hole array that enhances the pinning potential and
may result in a thermodynamic MI phase. Although this pic-
ture is correct for commensurate matching fields 共B/B
=1,3,4,7,..., for a triangular array of holes25兲,itisnot
strictly correct for incommensurate matching fields. At
B/B
=2, for example, there are two interstitial positions
with equal energy per unit cell. This would imply that hop-
ping of interstitials is possible also at incommensurate
matching fields, resulting in greater interstitial mobility, and
hence a weaker matching effect. This is not what we observe
in Fig. 2共c兲, where the fingers appear similar at the first and
second matching fields.
We now consider the theoretical plausibility of the multi-
quanta scenario. The condition given for multiple-quanta
pinning in holes is58 r3⬎
2for Ⰶd, where ris the hole
radius, dis the interhole distance, and =0/冑1−T/Tcis the
London penetration depth.4For r=0.2
m and 0
=0.15
m,59 we may expect multiple-quanta vortices for T
ⱕ88 K for our sample.
We now estimate the number of vortices pinned at a hole
as a function of temperature. We consider two physical pos-
sibilities, following Ref. 56. The first is that the nth vortex is
pinned as long as there is some force near the edge of the
hole that pushes it inward. This corresponds to a pinned state
MOTT INSULATOR PHASES AND FIRST-ORDER MELTING…PHYSICAL REVIEW B 79, 064523 共2009兲
064523-7
that may be metastable, depending on its free energy. The
saturation number
ns⬇r/2
共1兲
is the maximum nfor which this occurs. The second possi-
bility for multiple-quanta pinning is to require an equilibrium
pinned state, namely, the free energy of the nth vortex lo-
cated in the hole must be lower than its free energy far from
the hole. In this case, the number of pinned vortices is given
by56
n0⬇1
2ln r
2
冒ln 2
1.78r.共2兲
Both nsand n0decrease with T, which is consistent with the
observed downward steps in HIL共T兲. Substituting r
=0.2
m,
0=2 nm, 0=0.15
m, and Tc=90.5 K, we ob-
tain ns=1–21 for T=90.46 K down to T= 74.5 K and n0
=1–2 for T=85.2 K down to T= 76.1 K. Note that Eq. 共2兲
was derived from the pinning energy of a single hole. In the
case of an array of holes, one should compare the free en-
ergy of a pinned nth vortex to its free energy at the midpoint
between two neighboring holes. At this midpoint, there are
positive contributions to the free energy from the neighbor-
ing occupied holes. Thus a higher free energy of the pinned
vortex, or n⬎n0, would still be an equilibrium state of the
system. Equation 共2兲should therefore be considered as a
lower limit on the equilibrium number of vortices pinned to
a hole. The fact that interactions should raise estimated oc-
cupation numbers was also noted in Ref. 27.
We now compare these theoretical estimates to the experi-
mental values of nmax. We observed three decreasing steps in
TIL 关see Fig. 2共c兲兴, with T
˜
6⯝72 K, T
˜
5⯝75 K, and T
˜
4
⯝77 K. According to the schematic phase diagram plotted
in Fig. 8共d兲, the values of T
˜
nare lower than the temperatures
Tnat which the nth multiquanta vortex depins in the absence
of thermal fluctuations. From the schematic description in
Fig. 8共c兲, we see that the difference Tn−T
˜
nmay be estimated
from the difference in the TIL at, and slightly away from,
B/B
=n. From Figs. 6共c兲and 7共c兲, we estimate this
difference to be Tn−T
˜
n⬃2K.T6,T5, and T4are thus 74, 77,
and 79 K, respectively, or, equivalently, nmax共T⬍74兲=6,
nmax共74⬍T⬍77兲=5, nmax共77 ⬍T⬍79兲= 4, and nmax共T
⬎79兲=3. We find that the extracted nmax values are much
lower than the estimated nsand slightly higher than the esti-
mated n0. Thus the equilibrium multiquanta pinning scenario
described by Eq. 共2兲is more plausible. Note, however, that
Eqs. 共1兲and 共2兲are based on the assumption of fully pen-
etrating holes; the number of vortices trapped by surface
holes may be lower.60,61
Summarizing our discussion of the IL, the multiquanta
scenario provides a more consistent explanation as compared
to single-vortex pinning, both for the steps in the IL and for
the similar MI fingers at both commensurate and incommen-
surate matching fields. Indeed, recent simulations of BSCCO
with surface holes similar to those in the experiment62 indi-
cate that multiple-quanta occupation of surface holes does
occur, and that depinning may occur directly from holes.
Finally, we address the apparent contradiction of observ-
ing a FOT within the patterned regions at matching fields
B/B
=1 and 2, as shown in Figs. 6and 7. The observed
FOT occurs at the same field and temperature values as the
pristine solid-liquid transition Tm, and we therefore assume
that both transitions are of similar nature. This, however,
seems to contradict the existence of MI phases at matching
fields since in the MI phase the vortex lattice is ordered,
pinned, and incompressible up to temperatures well above
the pristine Tm.
The observed FOT can be understood qualitatively by tak-
ing the bulk beneath the surface holes into account. The sur-
face holes are only ⬃1.4
m deep, whereas the sample is
30
m thick. Although the depth at which vortices are still
sensitive to surface patterning is not known exactly, mag-
netic decorations of BSCCO crystals with square Fe periodic
surface patterns indicate that the hexagonal structure of the
lattice is recovered fully just 4.5
m beneath the pinning
potential at low temperatures.63 Thus one may expect that,
sufficiently deep below the upper surface, the vortex “tails”
will undergo a first-order melting transition at the pristine
Tm. At matching fields, however, the “tips” of the vortices at
the surface are pinned to the periodic surface holes and there-
fore behave as an incompressible solid. The resulting situa-
tion immediately above Tmis rather unique: the vortex tips
are in a solid MI state, while the vortex tails are liquid. At the
FOT, the vortex density in the bulk increases by ⌬B/
0,
where ⌬Bis the typical step in Bat Tm. If the tips of the
vortices were fully incompressible, this ⌬Bin the bulk would
be completely shielded and unobservable at the top surface.
Our data indicate, however, that within our experimental
range of parameters the compressibility is high, but finite.
Hence the MI top layer is sufficiently transparent to allow
observation of the paramagnetic peak at the FOT of the un-
derlying vortex tails. We conclude that upon increasing tem-
perature, the FOT observed within the patterned region at
matching fields indicates a transition from a solid bulk with
an incompressible solid surface to a unique state of an essen-
tially incompressible solid crust of vortex tips concealing a
vortex liquid in the bulk.
It is interesting to note that this solid crust could be used
as a tool to study the interesting possibility of surface melt-
ing that was suggested to exist in the vortex lattice,40,64 simi-
lar to surface wetting in atomic solids. This will require in-
vestigating crystals of different thicknesses and samples
patterned with holes at both the top and bottom surfaces.
No additional FOTs were observed, even in the presence
of shaking. This indicates that the depinning line of the in-
terstitial vortices, and the delocalization line of the vortices
pinned to holes, are apparently not FOTs. More accurate
magnetization measurements in the presence of shaking are
needed to check for the existence of second-order thermody-
namic transitions.
V. CONCLUSIONS
BSCCO crystals with arrays of surface holes were mea-
sured using differential magneto-optics accompanied by an
in-plane shaking field. We observe reduced dB /dH of the
patterned regions at integer matching fields B=nB
. These
GOLDBERG et al. PHYSICAL REVIEW B 79, 064523 共2009兲
064523-8
features are extremely narrow, with a width in Hof less than
1 Oe. The region of reduced dB/dH extends up to 87 K for
the first matching field, terminating 3.5 K below Tc. Shaking
allowed the equilibrium matching feature to be observed
both above and below the pristine melting line of the sample.
This observation is in contrast to previous dynamic measure-
ments of BSCCO crystals with periodic surface holes,35,38 in
which matching effects were observed only well above the
pristine melting line. Our finding of a sharply suppressed
compressibility of an equilibrated vortex lattice at the match-
ing fields is strong evidence of the existence of Mott insula-
tor phases.
We observe a first-order melting transition within the pat-
terned areas, both away from matching and at the first and
second matching fields. Surprisingly, this transition is not
shifted with respect to the pristine melting transition. We
interpret this transition as first-order melting of the vortices
in the pristine bulk beneath the patterned surface that results
in a step in vortex density. The added vortex tails beneath the
patterned surface force vortex tips through the surface, even
in regions of reduced compressibility. We emphasize that this
is an extremely unusual situation, in which the vortex tails
located in the bulk are in a liquid phase, while their tips
located near the surface are in a pinned, ordered phase, and
therefore solid.
The irreversibility line of the patterned regions is found to
be shifted upward in Hand Tto above the pristine melting
line Tmand displays steplike behavior, with almost no tem-
perature dependence between matching fields B=nB
. These
steps are consistent with a multivortex pinning scenario, with
an estimated maximum value of nmax=6 flux quanta pinned
to each surface hole at H/B
ⲏ5 and Tⱕ72 K.
Applied shaking shifts the irreversibility line to lower
field and temperature, enabling the observation of the first-
order transition. However, no first-order transitions related to
the surface holes, corresponding to depinning lines of inter-
stitial vortices, or delocalization of vortices pinned to holes,
were observed. Further transport measurements are neces-
sary to determine if an additional delocalization line exists at
higher temperatures, in analogy to the delocalization line of
the vortices residing on columnar defects that separates the
interstitial liquid from a homogeneous liquid.12
ACKNOWLEDGMENTS
We wish to thank A. Lahav for technical assistance. We
are grateful to Y. Goldschmidt and M. Opferman for provid-
ing insights from simulations. We thank V. B. Geshkenbein,
E. H. Brandt, and G. P. Mikitik for helpful discussions. This
work was supported by the U.S.-Israel Binational Science
Foundation 共BSF兲, by the Minerva Foundation, and by
Grant-in-aid from the Ministry of Education, Culture, Sport,
Science and Technology, Japan. E.Z. acknowledges the sup-
port of the German Israeli Foundation 共GIF兲.
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