Physica C 357±360 (2001) 446±449
www.elsevier.com/locate/physc
Vortex phase diagrams of Bi2Sr2CaCu2O8‡y in tilted
®elds studied by a Hall probe
M. Tokunaga *, S. Koya, M. Kishi, N. Kameda, K. Itaka, T. Tamegai
Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Received 16 October 2000; accepted 23 December 2000
Abstract
We studied the vortex phase diagram of Bi2 Sr2 CaCu2 O8‡y in tilted ®elds with respect to the c-axis of the crystal
through magnetization measurements by a micro Hall probe. The linear decrease of c-axis component of the melting
®elds, and second peak ®elds at low temperatures, against the in-plane component breaks down in the limit of both H kc
and H kab. Through the temperature and anisotropy dependence of the phase diagram, we discuss the possible origin of
the deviation from the linear dependence. Ó 2001 Elsevier Science B.V. All rights reserved.
PACS: 74.60.Ec; 74.72.Hs
Keywords: HTSC; Vortex; Phase diagram; Angular dependence
1. Introduction
In anisotropic superconductors, vortex states
under tilted ®elds have attracted considerable attention in the last decade. In highly anisotropic
Bi2 Sr2 CaCu2 O8‡y (BSCCO), ®eld of the ®rst
order transition of the vortex lattice [1] (H FOT ),
and the second peak ®eld (H p ) at low temperatures, depends on the angle between magnetic ®eld
and the c-axis (h) as H FOT h† ˆ H FOT 0† cos h ‡
a T † sin h† 1 [2,3]. This dependence is di€erent
from the scaling relation deduced from anisotropic
Ginzburg±Landau theory [4]. Thereby, some intrinsic physics in vortex matter should be considered to account for this angular dependence.
*
Corresponding author. Fax: +81-3-5841-6848.
E-mail address: mtokunaga@ap.t.u-tokyo.ac.jp (M. Tokunaga).
Theoretically, Koshelev explained the linear dependence of the c-axis component of H FOT on the
in-plane ®eld considering the combined lattice
ground state of pancake vortices (PVs) and Josephson vortices (JVs) [5]. According to his theory,
such a relationship breaks down at both limits of
in-plane component Hx ˆ 0 and 1. In the limit of
Hx ˆ 0, the linear dependence terminates when a
tilted lattice state takes over the combined lattice
state. On the other hand, the linear dependence
terminates where the cores of JVs overlap with
each other. Experimentally, recent AC susceptibility [6,7], transport [8,9], and magnetization
studies [10] revealed that the deviation from linear
relation when the magnetic ®eld is applied almost
parallel to the ab-plane. Moreover, Mirkovic et al.
reported novel stepwise behavior in this limit
through their results on transport measurements
[8,9]. On the other hand, as for low angle limit,
there is no systematic studies to date.
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 0 2 7 1 - 4
M. Tokunaga et al. / Physica C 357±360 (2001) 446±449
447
2. Experimental
We have carried out magnetization measurements on BSCCO crystals by a rotatable microHall-probe. For measurements at around h ˆ 0°,
we utilized a vector magnet. Measurements at
higher angles were performed by sweeping ®elds at
constant angles. Crystals of BSCCO were grown
by the traveling solvent ¯oating zone method. For
a comparative purpose, excessively oxidized and
Pb-doped (Bi2:2 x Pbx Sr1:8 CaCu2 O8‡y with x ˆ 0,
0.2, and 0.4) samples, which are both believed to
reduce the anisotropy, were prepared.
3. Results and discussion
Fig. 1(a) shows the c-axis components of transition ®elds (HzFOT and Hzp ) in optimally-doped
BSCCO as a function of in-plane ®eld (Hx ) at
various temperatures. Horizontal and vertical axes
are normalized by H FOT and H p at Hx ˆ 0 (H0 ) for
each temperature. The hatched rectangle in the
inset indicates the corresponding area at 60 K in
the main panel. When Hx =H0 > 0:2, the c-axis
components decrease linearly with the increase of
Hx . At the limit of h ˆ 0°, observed HzFOT and Hzp
deviate from the linear extrapolation of the data
from higher angles (dashed lines). We de®ne the
characteristic angle h0 as shown by arrows in the
®gure.
Temperature dependence of h0 for two crystals
of BSCCO (#1 and #2) are plotted by closed
symbols in Fig. 1(b). Open symbols in Fig. 1(b) are
the results for Pb-doped samples. According to
Koshelev, characteristic angle where the ground
state changes from tilted to combined lattice is,
h0 ˆ
6:8 4pk2 p cz
Bk Bz C ln :
c U0
s
1†
Here, c, s, and cz denote the anisotropy parameter,
interlayer spacing, and average spacing of JVs
along the c-axis, respectively. Bz is the c-axis
component of magnetic induction. C represents
the suppression of the e€ective Josephson energy
due to thermal motion of the PVs. Bk is a characteristic induction de®ned in Ref. [5]. Taking
c ˆ 300, k ˆ 200 nm= 1
T =Tc †2 †1=2 , and Bz ˆ
Fig. 1. (a) H FOT and H p at various temperatures in a Hz vs. Hx
plane. Horizontal and vertical axes are normalized by transition
®elds at Hx ˆ 0 (H0 ) for each temperature. Dashed lines are
guides to the eye. Description of arrows is in the text. The
hatched area in the inset indicates the corresponding area of the
main panel at 60 K. (b) Temperature dependence of h0 . Closed
and open symbols are the results for two BSCCO and Pb-doped
samples, respectively. The dashed line is calculated h0 by Eq. (1)
with c ˆ 300.
140, h0 at 60 K is estimated to be 8.3° for C ˆ 0:7.
This value is in reasonable agreement with our
results. The tendency that h0 increases with Pb
substitution is also consistent with the c dependence in Eq. (1). However, temperature dependence of the h0 seems not to be explained by the
calculation (the dashed line).
For higher angles, the H FOT and H p at various
temperatures from 50 to 82 K are plotted in a Hz
vs. Hx plane (Fig. 2(a)). The linear decrease of
HzFOT becomes less sensitive to Hx at higher angles.
The break points at various temperatures are
linked by a dashed line. As shown in Fig. 2(a),
448
M. Tokunaga et al. / Physica C 357±360 (2001) 446±449
Fig. 3. Temperature dependence of Hx for optimal and
overdoped BSCCO (closed symbols). Open symbols are the
ratio of average spacing of PVs and JVs in the ab-plane. For the
estimation of cy , we used c ˆ 300 and 150 for optimal and
overdoped BSCCO, respectively.
Fig. 2. (a) In-plane ®eld dependence of H FOT at higher angles.
The dashed line represents the break point of the linear dependence. (b) A normalized phase diagram with respect to the
c-axis (Hz ) and in-plane (Hx ) components of the break point at
each temperature. Arrows indicate the additional peak ®elds
observed at 27.5 K.
both out-of and in-plane components of this break
point (Hz and Hx ) increase considerably as temperature decreases. Fig. 2(b) is a normalized phase
diagram with respect to Hz and Hx . This ®gure
demonstrates that the results at all temperatures
roughly follow universal relation between out-of
and in-plane components. The similarity of the
phase diagram at each temperature suggests that
the ratio of the number of PVs and JVs seems to
be intrinsic to this anomaly. In the magnetization
curve at 27.5 K, additional peaks at lower Hz are
observed at the points marked by arrows. Possible
origins of this anomaly are discussed in Refs.
[6,9,10]. Further investigations are needed to clarify
the relation between this anomaly and the break
points.
Fig. 3 shows the temperature dependence of
the Hx (closed symbols). Results of a slightly
overdoped sample (Hp ˆ 505 Oe at 27.5 K) is
shown together. In both samples, Hx monotonically increases as temperature decreases. This result indicates that this anomaly may not be related
to the matching e€ect of JVs with interlayer
spacing, since it should be independent of temperature. Starting from the concept of combined
lattice, relative number of PVs against JVs decreases as the angle increases. Closed diamonds in
Fig. 3(b) represents the ratio between
average
pthe
 1=2
in-plane spacing of PVs a ˆ 2U0 = 3Bz † † and
1=2
that of JVs (cy ˆ ceff U0 =bBx † , b represents the
arrangement of JVs and ceff is the e€ective c in the
combined lattice state [5]). At Hx , the p
ratio
 becomes the order of unity for b ˆ 2= 3, and
remains almost constant with the changes of temperature and oxygen concentration. At higher
angles, where the linear in-plane ®eld dependence
of HzFOT and Hzp breaks down, matching e€ects
between PVs and JVs may be important.
4. Conclusion
In conclusion, we studied vortex phase diagram
in tilted ®elds by means of magnetization measurements. We observed deviation from the linear
dependence of the c-axis component against the inplane ®eld at the limits of small and large angles.
The deviation at small angles may be ascribed to
M. Tokunaga et al. / Physica C 357±360 (2001) 446±449
the change of ground state from combined to tilted
lattice. The deviation at higher angles occurs when
the average PV distance matches the average
spacing of JVs along the ab-plane.
Acknowledgements
This work is supported by CREST and Grantin-Aid for Scienti®c Research from the Ministry of
Education, Science, Sports and Culture of Japan.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
E. Zeldov, et al., Nature 375 (1995) 373.
B. Schmidt, et al., Phys. Rev. B 55 (1997) R8705.
S. Ooi, et al., Phys. Rev. Lett. 82 (1999) 4308.
G. Blatter, et al., Phys. Rev. Lett. 68 (1992) 875.
A.E. Koshelev, Phys. Rev. Lett. 83 (1999) 187.
M. Konczykowski, et al., cond-mat/9912284 (1999).
S. Nakaharai, et al., Phys. Rev. B 61 (2000) 3270.
J. Mirkovic, et al., Physica C 284±288 (2000) 733.
J. Mirkovic, et al., Phys. Rev. Lett. 86 (2001) 886.
S. Ooi, et al., Phys. Rev. B 63 (2001) 020501.
449