Table Of ContentMelting of ‘porous’ vortex matter
S. S. Banerjee1, A. Soibel1,∗, Y. Myasoedov1, M. Rappaport1, E. Zeldov1, M. Menghini2, Y. Fasano2, F. de la
Cruz2, C. J. van der Beek3, M. Konczykowski3, and T. Tamegai4
1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
2Instituto Balseiro and Centro At´omico Bariloche, CNEA, Av. Bustillo 9500, Bariloche, RN, Argentina
3
0 3Laboratoire des Solides Irradi´es, CNRS UMR 7642, Ecole Polytechnique, 91128 Palaiseau, France
0 4Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, and CREST Japan Science and
2 Technology Corporation (JST), Japan
(February 2, 2008)
n
a Bitter decoration and magneto-optical studies reveal that in heavy-ion irradiated superconduc-
J
tors, a ‘porous’ vortex matter is formed when vortices outnumber columnar defects (CDs). In this
4 state ordered vortex crystallites are embedded in the ‘pores’ of a rigid matrix of vortices pinned
1
on CDs. The crystallites melt through a first-order transition while the matrix remains solid. The
meltingtemperatureincreaseswithdensityofCDsandeventuallyturnsintoacontinuoustransition.
]
n At high temperatures a sharp kink in the melting line is found, signaling an abrupt change from
o crystallite melting tomelting of therigid matrix.
c
- PACS numbers: 74.60.Ec, 74.60.Ge, 74.80.-g, 74.72.Hs
r
p
u
s Melting of heterogeneous systems, and in particular of Tm and a sharp kink in the FOT line, which appar-
t. of nanocrystals embedded in porous rigid matrices, is entlyresultfromthecollapseofthematrixduetovortex
a
a complex process with many uncontrolled parameters. depinning from the CDs.
m
Metalandsemiconductornanocrystalswithfreesurfaces, The reported findings were obtained using Bitter dec-
-
d for example, usually show a decrease in their melting oration and differential magneto-optical (MO) [3] tech-
n temperature with decreasing size [1], whereas nanocrys- niques. High quality Bi2Sr2CaCu2O8 (BSCCO) crystals
o tals encapsulated in a porous matrix often display an (T ≈ 89 K) were covered by various patterned masks
c
c increase in melting temperature [2]. Although the con- and irradiated at GANIL by 1 GeV Pb ions with doses
[
tribution of the different factors is still a matter of de- corresponding to matching fields of B = 5, 10, 20, and
φ
1 bate,themeltingprocessisknowntodependonthesize, 50 G. Figure 1a shows schematically one of these masks
v
dimensionality, material properties of the nanocrystals whichresultsintheformationofCDsinBSCCOcrystals
7
and the matrix, as well as the interface energiesbetween only within the circular apertures of about 90 µm diam-
2
2 the materials [1,2]. In this workwe investigatean analo- eter. This patterning allows very sensitive simultaneous
1 gous,butamorecontrollablecompositesystem,whichis comparison of the vortex structure and the local melt-
0 a ‘porous’ vortex matter consisting of vortex nanocrys- ing processes in adjacent irradiated and pristine regions,
3
talsencapsulatedinamatrixofstronglypinnedvortices. which is not possible by other methods.
0
As shown below, this system is present in the commonly IntheabsenceofCDsthevorticesformtheBraggglass
/
t heavy-ion irradiated superconductors when the vortices phase[4]whichhasquasi-long-rangeorderwithno topo-
a
m outnumber the columnar defects (CDs). The rigid ma- logical defects, as seen in the pristine part of the mag-
trix is created by vortices localized on the network of netic decoration image in Figs. 1b and 1c. What hap-
-
d random CDs, while the softer nanocrystals are formed pens to this phase when sparse CDs are added? The
n within the ‘pores’ of this matrix by the interstitial vor- irradiated region in Fig. 1b shows that it is no longer
o
tices. The size of the nanocrystals can be readily varied Bragg glass since it has significant amount of topologi-
c
: fromseveralhundreddowntoafew vorticesbychanging cal defects (solid circles in Fig. 1c) and no orientational
v
the appliedfieldorthe densityofCDs. Wefindthatthis long-range order [5]. It is also not an amorphous or
i
X composite vortex matter reveals a number of intriguing glass phase in the usual sense, nor is it a simple poly-
r mechanisms: Similarly to the metallic nanocrystals in a crystal as discussed below. In the presence of CDs the
a
matrix, we observe for the first time a pronounced up- Bose glass (BG) theory [6–8] is usually applied, which
ward shift in the vortex melting temperature Tm, while describes the vortex matter in terms of anisotropic ho-
preserving thefirst-ordernatureofthetransition(FOT). mogeneously pinned medium. Such a description is ade-
With increasingdensity of CDs, the size of the pores de- quate for the commonsituation ofhigh irradiationdoses
creases, resulting in a larger shift in Tm. We also find a (Bφ > B), in which all the vortices reside on CDs and
critical point at which the FOT changes into a contin- the vortex pinning energies are comparable. In contrast,
uous melting. Moreover, the crystallites can melt while we investigate here mainly the opposite extreme of low
the matrix remains rigid. As a result, at high tempera- doses in which the vortices greatly outnumber the CDs,
tures we find an abrupt breakdown in the upward shift i.e.,B ≫B . Inthiscasethesystemisinherentlyhetero-
φ
1
geneous, consisting of two vortex populations with well apertures with B =20 G.
φ
separated characteristic energies: The vortices residing Inordertoinvestigatethephasetransitionoverawider
on CDs are strongly pinned and form a rigid network range of fields we have used differential MO imaging in
or matrix, whereas the interstitial vortices are localized which the field is modulated by 1 G (Fig. 3) instead
by significantly weaker elastic interactions and form rel- of the T modulation. In addition to detecting the FOT
atively soft crystallites within the pores of the matrix. [3],thismethodprovidesaverysensitivemeasurementof
In the following we refer to this state (upper parts of theirreversibilitylineataveryloweffectivefrequencyof
Figs. 1band1c)asporousvortexmatterinorderto em- about 0.1 Hz. Figure 3 demonstrates the determination
phasizetheimportantconsequencesoftheheterogeneous of the irreversibility line in the region where no FOT is
structure. present. In Fig. 3a the entire pristine sample is in the
Figure2 showsseveralframes froma ‘movie’[9]ofthe liquidstate,whiletheaperturesarestillsolid. Theaper-
melting process as a function of temperature T at two tures appear black, showing that the external field mod-
fields, 30 and 60 G. Each frame is obtained by taking ulation is shielded due to the enhanced pinning. Upon
the differencebetweentheMO imagesatT+0.15K and increasing the field the black apertures disappear (Figs.
T − 0.15 K and averaging a large number of such dif- 3b and 3c) revealing the value of the local irreversibility
ferential images, as described previously [3]. The bright field.
features show the regions in the sample that undergo a Figure 4 shows the location of the onset of the FOT
FOT within the temperature interval of 0.3 K at the in- for B =5, 10, 20, and 50 G obtained by T modulation
φ
dicated T, and the intensity of this bright paramagnetic (solid symbols) and of the irreversibility line obtained
signal is the equilibrium magnetization step ∆B at the by field modulation (open symbols). The solid lines are
transition [3,10]. Figure 2a shows the nucleation of the guides to the eye for the FOT lines which terminate at
liquid phase in the form of bright inclined strips in the thecorrespondingcriticalpoints. Theirreversibilitydata
central pristine region which arise from intrinsic sample coincide with the FOT line below the critical point and
disorder (see Ref. [3]). With increasing T (Fig. 2b) the smoothly extrapolate the location of the transition line
liquidexpands,remarkablyavoiding the irradiatedaper- to higher fields. The first interesting observation here is
tures. In Fig. 2c the entire central pristine part of the thatalthoughthestructuresofthe porousvortexmatter
sample isliquid, while the apertureswithB =20Gare and of the Bragg glass are very different (Fig. 1), their
φ
still solid. In Fig. 2d the central apertures melt at 82.45 phasediagramsforB =5GinFig. 4arealmostidenti-
φ
K,whichisabout1KaboveT oftheadjacentsurround- cal. The melting remainsFOT in mostof the field range
m
ing pristine regions in Fig. 2b. The apertures closer to and the melting line is shifted only slightly. This brings
the sample edges begin to melt in Fig. 2e. The FOT is us to an important conclusion that the quasi-long-range
equallystrongintheirradiatedandpristineregions: The order that characterizes the Bragg glass is not an essen-
∆B step derived from the paramagnetic melting signal tial requirementfor the existence of a FOT [11], and the
[10]isthesameinFigs. 2dand2b. Alsothewidthofthe presence of the short-range order within the crystallites
localmeltingtransitionisthesamewithinourresolution, is apparently sufficient.
i.e., each point in the sample melts within 0.3 K or less. We can understand the upwardshift in B (T) in Fig.
m
This is the first direct observation of an upward shift of 4 by generalizing the concept of the cage model. In a
the FOT by correlated disorder. Note that for small B pure system each vortex is confined in a potential cage
φ
the melting in the irradiated apertures occurs while the arising from the elastic interactions with its neighbors.
meltinginremotepristineregionsisstillinprogress,and Melting occurs when the transversethermal fluctuations
thereforethequantitativemeasurementoftheshiftinT of vortices hu i reach a certain fraction c of the lattice
m T L
has to be determined by comparing neighboring irradi- spacinga0. Theporesofthematrixareverticalcylinders
ated and pristine regions and cannotbe readily detected which provide an additional confining cage potential to
by global techniques. the interstitial vortices. This enhanced rigidity reduces
The melting process at 60 G (Fig. 2, second row) re- hu i, and hence stabilizes the solid crystallites within
T
veals two important differences. First, the shift of the the pores, thereby shifting B (T) upwards. The shift in
m
melting temperature, ∆T , is about 4 K (difference be- B (T) grows with B due to the decrease in the size of
m m φ
tweenFigs. 2iand2f) which is muchlargeras compared thepores. SincethepinningenergyoftheCDsisusually
to1Kat30G.InFig. 2h,forexample,theentirepristine significantlylargerthantheelasticenergy,themeltingof
samplehasmeltedwhiletheirradiatedaperturesarestill the crystallites mayoccur withoutthe destructionofthe
solid. Second, the brightness of the paramagnetic melt- matrix, which remains solid up to a higher temperature
ing signal ∆B in the apertures in Fig. 2i is much lower as described below. The rigid matrix may thus coexist
than in the pristine sample, and moreover, the melting with an interstitial liquid as shown theoretically [8] and
in eachaperture is broadenedover severalframes, as ex- in numerical simulations [12].
emplified in Fig. 2j. At still higher fields, above 100 G, Upon increasing the density of CDs the FOT is weak-
noparamagneticFOTsignalisdetectedintheirradiated enedanditeventuallytransformsintoacontinuoustran-
2
sitioninFig. 4. This transformationisseenmoreclearly to B = 50 G curve. In region 1 the crystallites in the
φ
by varying the temperature along a given B (T) line. poresarestabilizedbytherigidmatrixandmeltatBpor,
m m
The inset to Fig. 4, left axis, shows the height of the well above the pristine B (T). In region2 the pores are
m
equilibrium magnetization step ∆B vs. T for B = 20 liquid while the matrix remains intact, and hence Bpor
φ m
G. As the temperature is decreased∆B drops and even- reflects a melting transition of the softer of the two sub-
tually vanishes at a critical point (CP) below which no stancesinaheterogeneousmedium. Thisunconventional
discontinuity is found and the melting becomes contin- meltingcannotbedescribedbyasinglesetofparameters
uous. In addition, the width of the FOT, δT , shows since it depends on the properties of both the pores and
m
a significant broadening on approaching the CP (Fig. 4 the matrix, each described by a different set of param-
inset, right axis). This indicates that it is a true CP at eters with different field and temperature dependencies.
which the FOT transforms into a continuous transition As a result, Bpor(T) does not extrapolate to T as com-
m c
as expected theoretically [13]. This CP is unlike the ap- monly expected, but rather well above it. To the best of
parent point-disorder-induced CP in BSCCO at lower T our knowledge this is the first observation of a melting
where no appreciable broadening was observed [14], and or irreversibility line that has such an unusual property
where the FOT was found to be obscured by pinning- at high temperatures. This heterogeneous process col-
induced irreversibility in the liquid phase [15]. Here, in lapses sharply at T , abovewhich the matrix rapidly de-
k
contrast,the liquid is fully reversible and no such exper- localizes, resulting in a homogeneous liquid in region 3.
imental limitations exist. Figure 4 shows that the CP is The Bmtx line thus describes the melting process of the
m
shifted to lower fields with B [16]. It is interesting to matrix due to depinning from the CDs, which leads to
φ
notethatinYBCOtheCPwasobservedtoshiftupward immediate melting of the superheated crystallites. Thus
with CDs [17]. There, however, the CP is induced by incontrasttothecommonhomogeneousBGdescription,
the point disorder in the pristine crystals and the CDs the Bpor(T) and Bmtx(T) lines originate from two inde-
m m
just modify slightly its position by reducing the point- pendent heterogeneous processes, melting of crystallites
disorder-inducedvortexmeandering. Inourcase,incon- within a rigid matrix and the collapse of the matrix it-
trast,theCDsapparentlycreateanewcriticalpointthat self, resulting in a sharp kink at the intersection point.
does not exist in pristine crystals. We expect the Bmtx(T) line to extend also into the liq-
m
We now discuss another key finding which is a kink in uid phase [8], like the dashed line in Fig. 4, separating
the B (T) lines marked by the arrows in Fig. 4. This the interstitial liquid within a solid matrix in region 2
m
kinkbecomesveryprominentwhenthetemperatureshift from the homogeneous liquid 3. Our MO measurements
∆T between the irradiated and pristine melting lines cannot detect this line since both these regions are fully
m
is plotted as shown in Fig. 5. The inset displays the reversible. Inregion2,however,weexpectahigherc-axis
corresponding upward shift in field ∆B . In BG the- correlation than in region 3. Recent Josephson plasma
m
ory a kink in the transition line is expected to occur at resonance studies of BSCCO with low B indeed find
φ
B ≈B [7,8]. Wearguethatthekink inFig. 5is anew a recouping transition within the liquid phase at which
k φ
feature of completely different nature, which reflects the an enhancement in the interlayer phase coherence is ob-
collapse of the rigid matrix. The BG kink is experimen- served [19]. This extension of the Bmtx line should also
m
tally found to occur at B /B of 1/6 to 1 for large B , be detectable by transport measurements since the flux-
k φ φ
and B usually scales linearly with B [18,19]. The kink flow resistance of the interstitial liquid should be lower
k φ
in Fig. 5 occurs in the opposite regime of B /B ≈2 to thanofthehomogeneousliquid[8]. Thesestudieswillbe
k φ
8,anditdoesnotscalewithB . AthighB theBGkink the subject of future work.
φ φ
is a broad feature that was found to occur only along a In summary, when vortices outnumber CDs, hetero-
glass transition line [18,19]. In contrast, here the kink is geneity rather than the average properties of the lattice
extremely sharp, and it is the first observation of a kink has to be taken into account for a proper description
that occurs along a FOT line. Finally, the BG kink is a of the structure and the thermodynamic behavior of the
transition from a strong influence of the CDs below the vortex matter. We have found evidence for two mech-
kinktoamuchweakereffectofCDsabovethekink[7,8]. anisms: melting of superheated crystallites within the
Here, in contrast, the situation is just the opposite, as pores of a solid matrix and the destruction of the rigid
seen in the inset to Fig. 5. The effect of CDs is large matrix. The intersection point of these two independent
at fields above the kink, where the shift ∆B is almost processes results in a sharpkink in the observedmelting
m
constantandapproximatelyequaltoB ,anditcollapses line. The heterogeneousmelting canbe either first-order
φ
rapidly at T > T . This collapse occurs at T ≃ 75 K, orcontinuous,dependingontemperatureandthedensity
k k
withlittledependenceonB . Itisalsointerestingtonote of CDs. The porous vortex matter may thus provide a
φ
that for B ≤20 G the kink occurs along the FOT line, tunablemodelsystemforgeneralcomprehensionofmelt-
φ
while for B = 50 G it falls in the region of continuous ing of nanocrystals in porous materials.
φ
transition. ThisworkwassupportedbytheIsraelScienceFounda-
In Fig. 4 three regions are enumerated with respect tion and Center of Excellence Program,by the German-
3
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Germany,andbytheGrant-in-AidforScientificResearch 2398 (1992).
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ture,Japan. EZandFCacknowledgethesupportbythe 4666 (1995).
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Fundacion Antorchas - WIS collaboration program.
[9] Color movies of the melting process are available at
http://www.weizmann.ac.il/home/fnsup/. See AIP doc-
ument No.:.... for MO movies of vortex-lattice melting
at 30 G and 50
G.ThisdocumentmayberetrievedviatheEPAPShome
page(http://www.aip.org/pubservs/epaps.html)orfrom
ftp.aip.org in the directory /epaps/. See EPAPS home
*Currentaddress: BellLabs,LucentTechnologies,Mur-
page for more information.
ray Hill, NewJersey 07974.
[10] N. Morozov et al.,Phys. Rev.B 54, R3784 (1996).
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FIGURE CAPTIONS
Fig. 1. (a) Schematic of BSCCO crystal irradiated through a mask with an array of circular apertures. (b) Bitter
decoration image (B = 40 G, T = 4.2 K) showing pristine region at the bottom (Bragg glass) and a section of the
irradiated aperture (B =10 G) on top where porous vortex matter consisting of ordered crystallites embedded in a
φ
rigid matrix is formed. (c) Corresponding locations of six-fold (open circles) and of five- and seven-fold coordinated
vortices (solid circles) obtained by Delaunay triangulation.
Fig. 2. Melting process as a function of T at fields of 30 and 60 G in BSCCO crystal0.65×0.45×0.01 mm3 using
differential MO imaging with T modulation of 0.3 K. The bright regions are the areas that undergo melting within
the 0.3 K interval. (a) − (c) and (f) − (h): melting of the pristine regions while the irradiated apertures are still
solid. (d), (e),(i), and(j): melting ofthe irradiatedapertures(B =20G).Theareaoutsidethe crystalisblackened.
φ
Color movies of the melting process are available at http://www.weizmann.ac.il/home/fnsup/.
Fig. 3. DifferentialMOimagesusingfieldmodulationof1GinBSCCOcrystalofFig. 2atthreefieldsatT =68.4
K. (a) All the pristine regions are liquid while the irradiated apertures (B = 20 G) are still solid and irreversible,
φ
and hence appear black due to shielding. (b) Partial reversibility of the central apertures. (c) Central apertures are
liquid while the apertures closer to the edges begin to melt.
Fig. 4. The melting linesB (T)ofthe pristineandirradiatedregionswithindicatedB . Solid(open)symbolsare
m φ
temperature (field) modulation data showing the location of the FOT (irreversibility line). Solid (dotted) lines are
guides to the eyeofthe first-order(continuous)transitions thatterminate atthe criticalpoints J. Inset: The height
of the FOT equilibrium magnetization step ∆B which vanishes at the CP, and the local width δT of the FOT vs.
m
T.
Fig. 5. The shift ∆T in the melting temperature for different B with respect to the pristine T at various fields
m φ m
B. Inset: The upward shift ∆B in the melting field vs. T .
m m
4
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200 50 G
por
20 G
B
m
150
2
pristine
)
< 1
G
B = 5 G
f
(
10 G <
100 mtx
m B
m
B 100 B = 20 G
f
3
) 80 < 3
G
m d
60 T
2
50 ( m
B 40 (
K
D CP 1 )
20
T (K)
<
0
0
70 75 80 85
0
50 55 60 65 70 75 80 85 90
T (K)
m
Fig. 4
200
5 G
60
50 G
)
G
(
40
m T
k
150 B <
D 20 20 G <
10 G
) <
10 G
G <
5 G
0
(
50 60 70 80 90
<
T (K)
100 m
B
B
20 G k
<
50
B = 50 G
f
0
0 1 2 3 4 5 6 7 8 9 10 11
D
T (K)
m
Fig. 5
