PHYSICAL REVIEW B
VOLUME 58, NUMBER 9
1 SEPTEMBER 1998-I
In-plane anisotropy of vortex-lattice melting in large YBa2 Cu3 O7 single crystals
Takekazu Ishida and Kiichi Okuda
Department of Physics and Electronics, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan
Alexandre I. Rykov and Setsuko Tajima
SRL-ISTEC, 10-13 Shinonome 1-chome, Koto-ku, Tokyo 135, Japan
Ichiro Terasaki
Department of Applied Physics, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169, Japan
~Received 23 February 1998!
The vortex-lattice melting transition of a large untwinned YBa2 Cu3 O7 crystal (T c 591.6 K) in the case of
H i c axis appears as a jump in dc magnetization M dc at a melting transition T m and an abrupt transition in the
ac susceptibility x ac above T m . Similarly, M dc shows a jump at T m in the H i b and H i a configurations,
indicating that a first-order-melting transition occurs in a highly compressed vortex lattice. This is reinforced
by a lattice softening in x ac below T m and a sharp transition above T m . The melting lines fitted by H
5H 0 (12T/T c ) 1.5 give the relations H 0i a /H 0i c 57.2460.09, H 0i b /H 0i c 58.9360.12, and H 0i b /H 0i a 51.2360.01.
These are in excellent agreement with the anisotropy parameters g ca 5 Am c /m a 57.4460.25, g cb
5 Am c /m b 58.7960.21, and g ab 5 Am a /m b 51.1860.05. @S0163-1829~98!06733-2#
The electronic anisotropy of high-T c cuprates was first
recognized as uniaxial anisotropy or quasi-twodimensionality. The c-axis anisotropy of the untwinned
YBa2 Cu3 O7 was investigated by Farrell et al.1 They reported
the anisotropy parameter as g c 5 Am c / ^ m ab & 57.960.2 by
assuming an isotropic in-plane mass ^ m ab & . The theory also
gave a uniaxial torque formula by considering the interlayer
anisotropy g c . 2 However, Ishida et al.3 revealed three different anisotropy parameters of an untwinned YBa2 Cu3 O7 crystal as g ca 5 Am c /m a 57.4460.25, g cb 5 Am c /m b 58.79
60.21, and g ab 5 Am a /m b 51.1860.05 at 90 K from the
ab-dependent g c ( u ab ). There is an overwhelming difference
between g ca and g cb compared to the experimental error in
g c . In reality, we must deal with any anisotropy in
YBa2 Cu3 O7 in terms of the three dimensionality rather than
the quasi-two-dimensionality.
The vortex lattice melting of the first-order nature has
attracted considerable interest in recent years. Most of the
preceding works have concentrated on melting in a rather
isotropic lattice (H i c). Local-Hall-probe measurements detected an abrupt jump in the magnetization in highly twodimensional Bi2 Sr2 Ca1 Cu2 O8 in fields lower than 380 G.4
The entropy change DS per vortex per layer increased from
zero ~at the critical point! to 6k B as the temperature increased to the critical temperature T c . 5 The melting transition (H i c) has been reported using high-quality YBa2 Cu3 O7
crystals by means of the resistivity drop,6 the magnetization
jump,7,8 and the calorimetric measurement.9,10 Contrarily to
Bi2 Sr2 Ca1 Cu2 O8 , the first-order melting transition of the latent heat DS.0.7k B was evident in fields higher than 10 kG.
Computer simulation also revealed the first-order nature of
the vortex lattice melting when the field is parallel to the c
axis.11 It is still not clear whether a unified picture of the
melting transition over the wide range of dimensionality exists or not.
0163-1829/98/58~9!/5222~4!/$15.00
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Nelson12 first introduced the concept of the vortex entanglement in the mixed state of type-II superconductors.
Blatter et al.13 discussed a possible difference between the
disentangled liquid and the entangled liquid. The vortex disentanglement is supposed to increase the pinning capability
even in liquid. Ishida et al.14 carried out simultaneous measurements of the ac susceptibility and the dc magnetization
of untwinned YBa2 Cu3 O7 to clarify the dynamic nature of
vortex lattice melting (H i c). The first-order melting transition appeared as a jump in the dc magnetization while the
effectual pinning was probed by a sharp transition in x 8 and
a sharp peak in x 9 of width 0.2 K in 10 kG at temperatures
just above T m . In addition, precursor lattice softening phenomena were seen in x 8 and x 9 at temperatures below T m
down to 1 kG. The declination of the magnetic field by 5 °
from the c axis resulted in melting behavior similar to the
H i c case, denying the privilege status of the c axis to dominate the vortex lattice melting.
When the field is applied perpendicular to the c axis, the
vortex lattice is expected to deform extremely in the direction of the c axis. It is a challenging problem to reveal the
nature of a highly compressed vortex lattice (H'c). Transport studies15,16 suggested a melting transition for magnetic
field close to H i ab. The quasi-two-dimensional anisotropy
in latent heat was reported by Schilling et al.17 in the H i c
and H i a configurations. However, a computer simulation
discovered the absence of the first-order melting transition
for H'c. 18 To our knowledge, there are no studies on the
three-dimensional feature of the melting transition.
Thermodynamic measurements could see the phase transition with information on the latent heat. The ac susceptibility is a powerful tool for probing not only the phase transition but also the sequential change in vortex states upon
melting.3 In this paper, we describe the melting phase transition of an untwinned YBa2 Cu3 O7 crystal in the dc field
range of 0–50 kG by means of the ac susceptibility and the
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© 1998 The American Physical Society
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FIG. 1. The ac internal susceptibility (H ac53 G, f
5390 Hz) and the dc magnetization 4 p DM ~ZFC and FCC! of TF
No. 4, subtracted by a linear relation aT1b, as a function of temac
perature in 25 kG (H i c). T dc
m and T m denote the onset of the magnetization jump and the offset of the x 8 transition, respectively.
dc magnetization with emphasis on the three-dimensional anisotropy. We also discuss the nature of vortex liquid in
highly compressed vortex system.
Large single crystals of YBa2 Cu3 O72 d were grown by a
pulling technique.19 The samples were detwinned under
uniaxial pressure of 2–10 MPa in the flowing Ar gas at
600 °C, were cooled at a typical rate of 6 °C/h in the flowing
O2 gas, and were finally annealed at 490 °C in the flowing
O2 gas to ensure the oxygen content of 6.93 in the Lindemer
scale.20 The observation using a polarized microscope revealed the twin-free nature of the crystals. We used two
samples denoted TF No. 3 ~56.2 mg, l a 3l b 3l c .233
31.8 mm) and TF No. 4 ~50.9 mg, l a 3l b 3l c .233
31.5 mm) for ac and dc magnetic measurements. Our
samples benefit from the large l c and reveal the many-body
effect in the vortex matter in the H'c configurations.
The dc magnetization and the ac susceptibility ( x 81
2i x 19 ) were measured by a superconducting quantum interference device ~SQUID! magnetometer ~Quantum Design
MPMS-XL!. Both dc and ac fields were applied in parallel to
the a, b, or c axis in the form H dc1H acsin 2p ft. The
sample was mounted on a Delrin holder made by numericalcontrolled precision machining. The accuracy of the field
alignment is expected to be 62 °. The ac susceptibility is
presented by the internal susceptibility using the demagnetizing factors (N a 1N b 1N c 51). 21
In Fig. 1, we show x 8 and x 9 of TF No. 4 as a function of
decreasing temperature in 25 kG (H i c, f 5390Hz, H ac
53 G, N a .0.462). At first glance, a monotonic diamagnetic transition in x 8 and a single peak in x 9 seem not to
be informative but tedious. In Fig. 1, the relative change
4 p DM in the dc magnetization M of TF No. 4 is also shown
as a function of temperature in 25 kG in the zero-field-cooled
mode upon warming ~ZFC! as well as the field-cooled mode
upon cooling ~FCC!. The baseline DM dc50 is found for the
FCC data. One finds that the magnetization shows a clear
jump in the reversible temperature range of 84.7–84.9 K.
The onset of the melting transition of TF No. 4 at 84.9 K
almost coincides with the offset of the x 8 transition at 84.8 K
or of the x 9 peak as a function of decreasing temperature.
FIG. 2. Upper: the ac internal susceptibility (H ac53 G, f
5390 Hz) and the dc magnetization 4 p DM ~ZFC and FCC! of TF
No. 3, subtracted by a linear relation aT1b, as a function of temperature in 15 kG (H i a). Lower: 4 px (H ac51 G, f 5390 Hz)
and 4 p DM ~ZFC and FCC! of TF No. 3, subtracted by aT1b, in
30 kG (H i b).
The jump in magnetization is 0.34 G, to which the application of the Clausius-Clapeyron relation yielded the entropy
change DS50.97k B per unit-cell length of a single vortex.
We consider that the softening phenomena in x 8 ~Ref. 3! are
not seen because H ac is not large enough to fulfill the condition J c ;cH ac/4p l (a,b) during vortex lattice softening (J c ;
the critical current density!. The x 8 transition curve of TF
No. 4 shifts parallel to lower temperatures as H dc increases
when H i c. The ac susceptibility of TF No. 3 showed a behavior similar to TF No. 4 when H i c. Evidence for the lattice softening could not be seen in all cases because the
amplitude H ac was limited to 3 G by the instrument. We find
the melting temperature T ac
m (H i c) in the manner indicated in
Fig. 1.
In the upper part of Fig. 2, we show x 8 and x 9 of TF No.
3 in 15 kG as a function of decreasing temperature
(H i a, f 5390 Hz, H ac53 G, N a .0.368) and the
relative change 4 p DM in the dc magnetization 4 p M of TF
No. 3 in 15 kG as a function of temperature ~ZFC and FCC!.
In the lower part of Fig. 2, we show x 8 and x 9 of TF No. 3
in 30 kG as a function of decreasing temperature
(H i b, f 5390 Hz, H ac51 G, N b .0.222) and the
relative change 4 p DM in the dc magnetization M of TF No.
3 in 30 kG ~ZFC and FCC! is also shown. The susceptibility
profiles in the H i a and H i b cases are very similar to each
other. First, x 8 increases monotonically at lower temperatures due to the conventional bulk pinning. Then, x 8 decreases gradually arising from lattice softening and enhanced
pinning. Finally, x 8 shows a sharp transition in the narrow
regime. We interpret that vortex pinning is effectual in a
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FIG. 4. The melting lines of TF No. 3 for three different configurations (H i a, H i b, H i c) determined by the offset of the x 8
susceptibility (H ac53 G, f 5390 Hz).
FIG. 3. The ac internal susceptibility (H ac53 G, f
5390 Hz) of TF No. 3 as a function of temperature in various H dc
in the H i a and H i b configurations. The envelopes for the x 8 dips is
shown by the dashed parabolic line.
liquid state only in this limited temperature interval. The ac
susceptibility is in fact null at higher temperatures because
vortices stay in a truly free-liquid state. The contribution
from the eddy current and the paramagnetism is smaller than
the typical experimental resolution. The melting transitions
in the H i a as well as H i b cases are evidenced by the magnetization jumps. The magnetization jump 4 p DM reaches
0.32 G in H i a ~15 kG! and 0.29 G in H i b ~30 kG!. It is not
straightforward to compare the entropy jump DS with that in
H i c because there must be a limited number of pancake
vortices in H'c. The length of the vortex segment, which
carries the entropy change of k B , is denoted as DL. We
estimate DL H i a 50.46 nm in 15 kG, DL H i b 50.34 nm in 30
kG, and DL H i c 51.10 nm in 25 kG using the ClausiusClapeyron relation. This rules out the explanation that a possible misalignment is responsible for the melting behavior of
M dc , x 8 , and x 9 . The x ac and M dc profiles in H i a and H i b
are very similar to those reported for H i c. 3 The completion
of the melting transition coincides well with the offset of x 8
transition in the accuracy of 0.1 K, as was seen for the H i c
case.
In Fig. 3 we compare x 8 and x 9 ( f 5390 Hz, H ac
53 G) of TF No. 3 as a function of decreasing temperature
in various H dc for the H i a and H i b configurations. At
higher H dc , the melting temperature T m is defined as a dip
temperature in x 8 . At lower H dc than 1 kG, we read T m as a
crossing point of x 8 with a parabolic envelope line ~see
dashed lines!. This implicates a continuous contour of J c in
the H-T plane. The x 8 amplitude at T m in H i a is appreciably larger than that in H i b as indicated by the dashed and
dotted envelopes in the lower figure. We consider that the
relevant J c at T m contains the CuO chain contribution when
H i a. On the contrary, the CuO chains are not so effectual
that they shield the magnetic field when H i b. Another point
is that x 8 shows the diamagnetism at temperatures above 92
K when H dc50. This diamagnetism is more appreciable in
H i a than in H i b ~see the dotted transition curve in the lower
figure!, indicating the dominance of the CuO chains at above
92 K. It is possible that T c of the CuO chains is somewhat
higher than that of the CuO2 planes due to the lower dimensionality.
In Fig. 4, we show the melting lines of TF No. 3 for three
different configurations (H ac53 G, f 5390 Hz) at temperatures above 85 K. We fit the lines by H5H 0 (1
2T/T c ) n , where the exponent n is fixed as 1.5 for mutual
comparison of the prefactors since the anisotropy is not sensitive to the exact choice for the exponent n. We obtain H
5(9195638)(12T/T c ) 1.5, H5(11339676)(12T/T c ) 1.5,
and H5(1270614)(12T/T c ) 1.5 in kG units for H i a, H i b,
and H i c, respectively. The anisotropy parameters of an untwinned YBa2 Cu3 O7 crystal were reported as g ca 57.44
60.25, g cb 58.7960.21, and g ab 51.1860.05 at 90 K.3 It is
interesting to note that the relations among the prefactors
(H 0i a /H 0i c 57.2460.09,
H 0i b /H 0i c 58.9360.12,
H 0i b /H 0i a
51.2360.01) are in remarkable agreement with the three
anisotropy parameters.
A simple explanation for the correlation between the effective mass and the melting field is as follows. Blatter
et al.13 give the melting field expression as
B m. b m
c 4L
Gi
S D
H c2~ 0 ! 12
T
Tc
n
,
~1!
where b m .5, G i is the Ginzburg number, c L is a Lindemann number, H c2 (0) is the upper critical field at T50, and
n52 for the two-fluid model. A similar relation between the
melting field and the anisotropy is obtained by taking account of the electromagnetic coupling with n51.5.22 The
ic
(0)
upper critical field is very anisotropic as H c2
ia
H c2 (0)5 f 0 /2pj b (0) j c (0),
and
5 f 0 /2pj a (0) j b (0),
ib
(0)5 f 0 /2pj c (0) j a (0). Since the Ginzburg number G i
H c2
5 21 @ T c /H 2c (0) j a (0) j b (0) j c (0) # 2 is a measure of thermal
energy with respect to the minimal activation energy of the
superconducting condensation, it is essentially independent
of the field direction. It is reasonable to consider
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that the melting field H m reflects the anisotropy in the
upper critical field H c2 (0). Our results implicate
ib
ic
that the relations H 0i b /H 0i c .H c2
(0)/H c2
(0)5 j b (0)/ j c (0)
ia
ic
ia
ic
(0)5 j a (0)/
5 Am c /m b 5 g cb .8.8, H 0 /H 0 .H c2 (0)/H c2
ib
ia
ib
ia
j c (0) 5 Am c /m a 5 g ca .7.5, and H 0 /H 0 .H c2
(0)/H c2
(0)
A
5 j b (0)/ j a (0)5 m a /m b 5 g ab .1.2 are mainly responsible
for determining the melting phase boundaries. The results are
consistent with the melting line reported by Schilling et al.17
by means of latent heat in untwinned YBa2 Cu3 O7 in quasitwo-dimensional configurations (H i c and H i a).
One may argue with the influence of the intrinsic pinning
on the melting transition in H'c. However, the complete
absence of intrinsic pinning at temperatures around T c in
H'c was confirmed by torque measurements with an accuracy of 0.18°. 14,23 The melting lines of the H'c configura-
tions of Fig. 4 are located in the absent region of the TachikiTakahashi intrinsic pinning.24
In conclusion, the melting transition of YBa2 Cu3 O7 has a
complete three-dimensional nature. The melting transitions
were well evidenced by the ac susceptibility and the dc magnetization even in the highly compressed vortex configurations (H i a,H i b). It seems that the melting lines are entirely
governed by the three-dimensional mass anisotropy. The two
dimensionality is not essential to achieve the melting transition in high-T c cuprates. The liquid phase of the vortex matter in YBa2 Cu3 O7 can be separated into the narrow pinned
regime and the wide depinned regime.13
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2
15
We thank M. Yakabe for his help in measurements. This
work at OPU was partially supported by a Grant-in-Aid for
Scientific Research ~Project No. 08454100! granted by the
Ministry of Education, Science, and Culture of Japan. The
work at ISTEC-SRL was partially supported by NEDO.
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