I-V characteristics of the vortex state in MgB2 thin films
I-V characteristics of the vortex state in MgB2 thin films
Huan Yang,1 Ying Jia,1 Lei Shan,1 Yingzi Zhang1 and Hai-Hu Wen1∗
National Laboratory for Superconductivity, Institute of Physics and National Laboratory for Condensed Matter Physics,
Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, P. R. China
Chenggang Zhuang,2,3 Zikui Liu,4 Qi Li,2 Yi Cui2 and Xiaoxing Xi2,4
Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
Department of Physics, Peking University, Beijing 100871, PR China and
Department of Materials Science and Engineering,
The Pennsylvania State University, University Park, Pennsylvania 16802, USA
(Dated: October 22, 2018)
The current-voltage (I-V ) characteristics of various MgB2 films have been studied at different
magnetic fields parallel to c-axis. At fields µ0H between 0 and 5 T, vortex liquid-glass transitions
were found in the I-V isotherms. Consistently, the I-V curves measured at different temperatures
show a scaling behavior in the framework of quasi-two-dimension (quasi-2D) vortex glass theory.
However, at µ0H ≥ 5 T, a finite dissipation was observed down to the lowest temperature here, T =
1.7 K, and the I-V isotherms did not scale in terms of any known scaling law, of any dimensionality.
We suggest that this may be caused by a mixture of σ band vortices and π band quasiparticles.
Interestingly, the I-V curves at zero magnetic field can still be scaled according to the quasi-2D
vortex glass formalism, indicating an equivalent effect of self-field due to persistent current and
applied magnetic field.
PACS numbers: 74.70.Ad, 74.25.Qt, 74.25.Sv
I. INTRODUCTION
Since the discovery of the two-gap superconductor
MgB2 in 2001,
1 the mechanism of its superconductiv-
ity and vortex dynamics has attracted considerable in-
terests. The two three-dimension (3D) π bands and
two quasi-two-dimension (quasi-2D) σ bands in this sim-
ple binary compound seem to play an important role
in the superconductivity,2 as well as the normal state
properties.3,4 The two sets of bands have different en-
ergy gaps, i.e., about 7 meV for the σ bands, and about
2 meV for the π bands.5,6 And the coherent length of
the π bands is much larger than that of the σ bands2.
Many experiments have demonstrated that the π-band
superconductivity is induced from the σ-band and there
is a rich evidence for both the interband and intraband
scattering. Owing to the complicated nature of super-
conductivity in this system, its vortex dynamics may ex-
hibit some interesting or novel features. Among various
experimental methods, measuring the current-voltage (I-
V ) characteristics at different temperatures and magnetic
fields can provide important information for understand-
ing the physics of the vortex state. Up to now, the
transport properties of MgB2 have been studied on both
polycrystalline bulk samples7 and thin films8. In both
cases, the I-V characteristics demonstrated good agree-
ment with the 3D vortex glass (VG) theory. This was
partially due to the limited magnetic fields in the experi-
ment. In addition, it has been shown that the properties
of MgB2 are very sensitive to the impurities and defects
introduced in the process of sample preparation, and the
vortex dynamics must be influenced, too. Therefore, it
is necessary to investigate the vortex dynamics in high
quality MgB2 epitaxial thin films and to reveal the in-
trinsic properties of the vortex matter in this interesting
multiband system. In this paper, we present the I-V
characteristics of high-quality MgB2 thin films measured
at various temperatures and magnetic fields. The vortex
dynamics in this system is then investigated in detail.
II. EXPERIMENT
The high-quality MgB2 thin films studied in this work
were prepared by the hybrid physical-chemical vapor de-
position technique9 on (0001) 6H-SiC substrates. All the
films had c-axis orientation with the thickness of about
100 nm. Fig. 1 (a) shows the θ-2θ scan of the MgB2 film,
and the sharp (000l) peaks indicate the pure phase of the
c-axis orientation of MgB2. In order to show the good
crystallinity of the film, we present in Fig. 1(b) the same
data in a semilogarithmic scale which enlarges the data
in the region of small magnitude. It is clear that, besides
the background noise, we can only observe the diffraction
peaks from MgB2 and the SiC substrate, i.e., there is no
trace of the second phase in the film. The c-axis lattice
constant calculated from the MgB2 peak positions was
about 3.517 Å(bulk value1: 3.524 Å). The φ scan (az-
imuthal scan) shown elsewhere9 indicated well the six-
fold hexagonal symmetry of the MgB2 film matching the
substrate. The full width at half maximum (FWHM)
of the 0002 peak taken on the film in θ-2θ scan [MgB2
0002 peak in Fig. 1(a)] and ω scan [rocking curve, shown
in Fig. 1(c)] is 0.15◦ and 0.39◦, respectively. The scan-
ning electron microscopy (SEM) image in Fig. 1(d) gave
a rather smooth top surface view without any observable
http://arxiv.org/abs/0704.0126v2
2x103
4x103
6x103
24 26
3x103
6x103
10 20 30 40 50 60 70 80
 (degrees)
FWHM=0.39o
 (0002)
500nm
2  (degrees)
FIG. 1: (a) X-ray diffraction pattern of the MgB2 film on a
(0001) 6H-SiC substrate in the θ-2θ scan, which shows only
the 000l peaks of MgB2 in addition to substrate peaks, in-
dicating a phase-pure c-axis-oriented MgB2 film. (b) The
semilogarithmic plot of the θ-2θ scan. (c) The rocking curve
of the 0002 MgB2 peak, which shows the FWHM of about
0.39◦. (d) The SEM image of the MgB2 film, which shows
the smooth surface without obvious granularity.
grain boundaries, which suggested that the film had a
homogeneous quality. Ion etching was used to pattern a
four-lead bridge with the effective size of 380× 20 µm2.
The resistance measurements were made in an Oxford
cryogenic system Maglab-Exa-12 with magnetic field up
to 12 T. Magnetic field was applied along the c axis of
the film for all the measurements. The temperature sta-
bilization was better than 0.1% and the resolution of the
voltmeter was about 10 nV. We have done all the mea-
surements on several MgB2 films, and the experimental
data and scaling behaviors are similar; so, in this paper,
we present the data from one film.
In Fig. 2, we present the resistive transitions (R-T re-
lations) of a MgB2 thin film measured at various mag-
netic fields in a semilogarithmic scale. The current den-
sity in the measurement was about 500 A/cm
, much
smaller than the critical value for low temperatures,
106 A/cm
2 10. It can be determined from Fig. 2 that
the sample had a superconducting transition tempera-
ture of Tc = 40.05 K, with a transition width of about
0.5 K. Its normal state resistivity was about 2.45 µΩcm
and the residual resistance ratio [≡ ρ(300 K)/ρ(42 K)]
was about 6.4. The I-V curves were measured at various
temperatures for each field, and then we got the electric
0 10 20 30 40 50
H=0 T
H=1 T
H=3 T
H=6 T
T (K)
FIG. 2: Temperature dependence of resistive transitions for
µ0H = 0, 1, 3, and 6 T, with the current density j =
500 A/cm
field (E) and the current density (j) according to the
sample dimension. The current density was swept from
5 to 105 A/cm
during the I-V measurements.
III. THEORETICAL MODELS
In the mixed state of high-Tc superconductors with
randomly distributed pointlike pinning centers, a second-
order phase transition is predicted between VG state and
vortex-liquid state.11 The I-V curves at different tem-
peratures near the VG transition temperature Tg can be
scaled onto two different branches12 by the scaling law
j (T − Tg)
ν(z+2−D)
|T − Tg|
ν(D−1)
. (1)
The scaling parameter z has the value of 4–7, and ν ≈ 1–
2; D denotes the dimension of the system with the value
3 for 3D and 2 for quasi-2D13; f+ and f− represent the
functions for two sets of the branches above and below
Tg. Above Tg, the linear resistivity is given by
ρlin = dE/dj|j→0 ∝ (T − Tg)
ν(z+2−D)
. (2)
At Tg, the electric field versus the current density curve
satisfies the relationship
E(j)|T=Tg ≈ j
(z+1)/(D−1). (3)
In 2D superconductors at µ0H = 0 T, a Berezinskii-
Kosterlitz-Thouless (BKT) transition was found at a spe-
cific temperature TBKT.
14 At TBKT, E ∝ j
3, which is a
sign of the BKT transition. A continuous change from
the BKT transition at zero field to a quasi-2D VG transi-
tion, and then to a true 2D VG transition with Tg = 0 K
was found in TlBaCaCuO film,15 which shows a field-
induced crossover of criticalities.
A 2D VG transition may exist in a true 2D system
with Tg = 0 K, i.e., there is no zero-resistance state at
any finite temperatures. The E-j curves can be scaled
T 1+ν2D
, (4)
where T0 is a characteristic temperature, ν2D ≈ 2, and
p ≥ 1, while g is a scaling function for all temperatures
at a given magnetic field. The linear resistance is given
ρlin ∝ exp[−(T0/T )
p]. (5)
This 2D scaling law can be achieved in the very thin
films17 or in highly anisotropic systems at high magnetic
fields.18,19
IV. EXPERIMENTAL RESULTS AND
DISCUSSIONS
A. Quasi-two-dimension vortex-glass scaling in the
low-field region (µ0H < 5 T)
The E-j characteristics have been measured at various
magnetic fields up to 12 T. In Fig. 3 we show the typical
example at µ0H = 1 T for (a) E-j curves and (b) the cor-
responding ρ-j curves in double-logarithmic scales. It is
obvious that when the temperature goes below some par-
ticular value (this is actually the vortex-glass transition
temperature Tg according to following discussions), the
resistivity falls rapidly with decreasing current density
and finally reaches the zero-resistance state which is the
characteristic of the so-called VG state. At the tempera-
tures above Tg, the resistivity remains constant in small
current limit. The current density of 500 A/cm
used in
ρ-T measurement shown in Fig. 2 lies in this linear resis-
tivity regime from about 10−3 to 1 µΩcm. Consequently,
these data sets provide the basic information on scaling
if the data are describable by the VG theory.
The inset in Fig. 4 shows the data of the ρlin versus
(T −Tg) and the fit to Eq. (2). The data are the same as
those shown in Fig. 2 for µ0H = 1 T, and the attempt Tg
value is 31.4 K. In this double-logarithmic plot, the slope
of the linear fitting gives just the exponent of ν(z+2−D),
and the determined value is 8.08± 0.05. In order to have
reasonable values for ν and z, the dimension parameterD
needs to be chosen as 2, i.e., the investigated system has
the property of quasi-2D, which is similar to the situation
found in BiSrCaCuO.13,20 This is further supported by
the VG scaling of the data at 1 T. As shown in the main
frame of Fig. 4, the scaling experimental E-j curves form
two universal branches corresponding to the data above
and below Tg (31.4 K) with ν = 1.32 and z = 6.12.
At very large current density or a temperature near the
onset of superconducting transition, the free flux flow
regime dominates and, hence, the data do not scale. The
101 102 103 104 105
10 nV
j (A/cm2)
H=1 T
FIG. 3: (Color online) (a) E-j characteristics measured at
fixed temperatures ranging from 30 to 36 K for µ0H = 1 T.
The increments are 0.30 K in the range from 30.00 to 31.20 K,
and 0.25 K in the range from 31.50 to 34.00 K respectively,
and finally 35 K on the top. The dashed line shows the po-
sition of Tg, and the symbols denote the segments that scale
well according to the quasi-2D VG theory. The thin solid
lines denote also the measured data, however, located outside
the scalable range. (b) ρ-j curves corresponding to the E-j
data in (a). The thick solid line in (b) denotes the voltage
resolution of 10 nV.
1 2 3 4 5 6
 From R(T)
 From E(j)
100 101 102 103 104 105 106
Quasi-2D VG Scaling for 
H=1 T
D=2, T
=31.4 K, =1.32, z=6.12
j/|T-T
| (D-1)
FIG. 4: (Color online) Quasi-2D VG scaling of the E-j curves
measured at 1 T. The inset shows a double-logarithmic plot
of the temperature dependence of the linear resistivity. The
dashed line is a guide for the eyes.
3 6 9 121518
 From R(T)
 From E(j)
10-2 10-1 100 101 102 103 104 105
10-10
Quasi-2D VG Scaling for H=3 T
D=2, T
=15.4 K, =1.00, z=7.70
j/(T|T-T
| (D-1))
10-1 100 101 102 103 104 105 106
10-11
Quasi-2D VG Scaling for H=3 T
D=2, T
=15.4 K, =1.17, z=6.58
j/|T-T
| (D-1)
FIG. 5: (Color online)(a) Scaling curves of the E-j data mea-
sured in 3 T based on the quasi-2D VG scaling theory. The
inset shows a log-log plot of the temperature dependence of
the linear resistivity. (b) VG scaling with another form of
scaling variable j/(T |T − Tg|
ν(D−1)
symbols in the figure denote the range of the data well
described by the scaling law.
The situation at µ0H = 3 T is similar to that at
µ0H = 1 T. As shown in Fig. 5(a), the determined pa-
rameters are Tg = 15.4 K, ν = 1.17, and z = 6.58. Inter-
estingly, the previous work on MgB2 film
8 indicated that
the 3D VG scaling theory (D = 3) is a better choice in
describing the I-V characteristics in this system, though
this experiment was done at magnetic fields lower than
1 T. Moreover, the I-V curves were demonstrated to scale
well by using the argument of j/(T |T − Tg|
). The same
conclusions were also drawn on the polycrystalline MgB2
samples7. In order to clarify this issue, we also analyzed
our data using the form suggested in Ref. 8. As shown in
Fig. 5(b), such a scaling with j/(T |T − Tg|
ν(D−1)
) as the
scaling variable is worse than that with j/ |T − Tg|
ν(D−1)
Most importantly, the dimension parameter D is still re-
quired to be 2 instead of 3 as proposed in Refs. 7 and
8. This confusion can be easily understood in terms of
the two-band superconductivity of MgB2. As we know,
there are two types of bands contributing to the super-
conductivity of MgB2, namely, the 3D π bands and the
101 102 103 104 105
H=6 T
j (A/cm2)
FIG. 6: (Color online) ρ-j data at temperatures 1.7 K and
4 K to 20 K with 2 K-step, for µ0H = 6 T. Temperature of
the isotherms increases from bottom to top.
2D σ bands. Therefore, the structure of the vortex mat-
ter must be affected by both of them. Although the su-
perconductivity of π bands, induced possibly by that of
σ bands, is much weaker, it provides a large coherence
length with 3D characteristics in the low-field region.
Therefore, the vortices in this system may be quasi-2D
like and, at the same time, they can possess large cores
characterized by the coherence length of the π band su-
perfluid. In this sense, the quasi-2D scaling should be
more appropriate than the 3D one. However, when a
higher disorder is induced in the system, especially in
the boron sites, the interband scattering gets stronger
and the anisotropy decreases, which may lead to a 3D
vortex scaling. In this case, a more rigid vortex line can
be observed, especially at low fields.21 The good quasi-2D
scaling at 1 and 3 T demonstrated here suggests that the
phase transition from VG to the vortex liquid in MgB2
resembles that in the high-Tc superconductors. Together
with the data shown below, we can safely conclude that
a vortex glass state with zero linear resistivity can be
achieved in the low field region due to the presence of the
finite superfluid density from the π bands. Regarding the
VG scaling22 a principal requirement is a proper deter-
mination of Tg, namely the temperature with a straight
logE-log j curve in the low dissipation part. The toler-
ance for Tg variation is very small (about ±0.3 K). With
an inappropriately chosen Tg, the scaling quality dra-
matically deteriorates and, simultaneously, the values of
ν and z quickly deviate from those reported above and
those proposed by theory. This validates our analysis
here.
10-3 10-1 101 103
A scaling attempt with quasi-2D VG for 
H=6 T
D=2, T
=0 K, =3, z=0.58
j/|T-T
| (D-1)
 Qusi-2D VG theory
 From R(T)
 From E(j)
FIG. 7: (Color online) The scaling of the E-j isotherms with
quasi-2D VG model for µ0H = 6 T. The inset shows the
deviation of the ρlin vs T − Tg (Tg = 0) relation from the
linearity in the double logarithmic scale.
B. Anomalous vortex properties in high field
region
As shown in Fig. 2, when the magnetic field reaches
6 T, no zero-resistance state can be observed down to
the lowest temperature, here 1.7 K. Consequently, no
VG transition exists above 1.7 K at this field, as shown in
Fig. 6. The shape of the curve at T = 1.7 K suggests that
the resistivity goes to a finite value as the current den-
sity approaches zero.23 As shown in Fig. 7, the ρlin versus
T − Tg seriously deviates from linearity for any possible
Tg value, indicating the inapplicability of Eq. (2) in the
present case. Correspondingly, the quasi-2D scaling law
fails here. A natural explanation is that, with increas-
ing field, the 3D supercurrent from π bands is seriously
suppressed6,24 and the quasi-2D vortex structure trans-
forms into a 2D-like one dominated by the σ band super-
fluid. In Fig. 8, we show our attempt to apply 2D VG
scaling on the data. Surprisingly, this attempt also failed,
even though this model has been successfully applied to
the layered superconductors with large anisotropy (or 2D
property) such as Tl- and Bi-based high-Tc thin films at
high magnetic fields.18,19
The most reasonable explanation for this anomaly is
that the supercurrent contribution from the π bands is
much easier to suppress by the magnetic field than that
from the σ bands, since the gap in the π bands is several
times smaller than that in the σ band. We suggest that
at high magnetic fields (above 5 T), a different vortex
matter state is formed, composed of quasi-particles from
the π bands and vortices formed mainly by the residual
superfluid from the σ band. The π-band quasiparticles
diminish the long range phase coherence of the supercon-
ducting phase, which leads to a finite dissipation. Once
the long range superconducting phase coherence is de-
stroyed by the proliferation of a large amount of these
5 10 15 20
10-4 10-3 10-2 10-1 100 101 102 103
A scaling attempt with 2D VG for 
H=6 T
=2, p=1, T
=20 K
j/T1+ 2D
 2D VG theory
 From R(T)
 From E(j)
T (K)
FIG. 8: (Color online) Attempted scaling of the data with
2D VG model [Eq. (4)] for µ0H = 6 T. The inset shows the
nonlinearity of the relationship between ρlin and temperature,
the solid line shows the theoretical curve of true 2D VG theory
[Eq. (5)].
π-band quasiparticles, neither 3D nor quasi-2D VG scal-
ing is applicable. Such a mixed state is obviously diffi-
cult to be simply described by any known scaling theory.
Recently, scanning tunneling microscopy studies showed
that the quasiparticles of the π bands disperse over all of
the superconductor, both within and outside the vortex
cores25, which strongly supports our arguments. This is
the basis for the explanation of the nonvanishing vortex
dissipation at high magnetic fields in a zero temperature
limit found recently on MgB2 thin films.
C. Self-field effect at µ0H = 0 T
For a 2D layered superconductor in zerofield, the above
mentioned BKT transition may exist and be reflected
in the I-V characteristics15. In the present MgB2 sam-
ples, we have not found any evidence of this transition
in low magnetic fields which would be consistent with
the quasi-2D (instead of 2D) configuration of the vor-
tex matter. Moreover, both the E-j curves and the ρ-j
curves (as presented in Fig. 9) are similar to the situation
of µ0H = 1 T. Considering the narrow transition width
at zero-field, we did the measurement carefully with an
increment of 0.05 K. Obviously, there is no E(j) curve
which satisfies the E ∝ j3 dependence, as expected by the
BKT theory. Since the current can induce self-generated
vortices, it might be interesting to look at whether the
quasi-2D VG model applies here.
Similar in Sec. IV A, we present ρlin versus (T − Tg)
in a double logarithmic plot. From this graph, we de-
termined the exponent in Eq. 2 (as shown by the inset
of Fig. 10(a)). A good quasi-2D scaling was obtained
with parameters Tg = 39.94 K, ν = 1.12, and z = 6.61,
as presented in Fig. 10(a). Using the parameters deter-
) 0H=0 T
101 102 103 104 105
j (A/cm2)
10 nV
FIG. 9: (Color online) (a) E-j data at various temperatures
from 39.7 K to 40.5 K with an interval of 0.05 K for µ0H =
0 T, the symbols denote the region, where the data are scaled
(from 39.70 K to 40.30 K). Temperature of the isotherms
increases from bottom to top. The dashed line shows the
position of Tg and the symbols denote the segments, which
scale well according to the quasi-2D VG theory. The thin solid
lines are also the measured data lying outside the scalable
range. (b) ρ-j curves corresponding to the E-j data in (a).
The thick solid line in (b) denotes the voltage resolution of
10 nV.
mined here, one finds a self-consistency with the value
of ν(z + 2 − D), as determined in fitting the linear re-
sistivity [Eq. (2)]. Both the temperature dependence of
ρlin and the scaling curves at µ0H = 0 T are similar
to the situation at small field µ0H = 0.1 T [shown in
Fig. 10(b)] and µ0H = 0.5 T (not shown in this pa-
per), except for the slight differences of the scaling pa-
rameters. The scaling parameters including the ones at
µ0H = 0.5 T are listed in Table I. It was proven that
current and magnetic field exhibit analogous effects in
suppressing superconductivity and generating quasipar-
ticles in conventional superconductors.26 Similarly, the
current-induced self-field may lead to a similar effect in
the vortex state as an applied magnetic field. Nonethe-
less, the good agreement of this simple scaling law with
the zero-field data is interesting and worth studying in
detail. Moreover, the values of ν and z for zero field are
very close to those for µ0H = 1 and 3 T, indicating a
similar vortex dynamics in the whole low-field region.
H=0 T
101 102 103 104 105
j (A/cm2)
10 nV
FIG. 10: (Color online) (a) Quasi-2D VG scaling of the data
measured at 0 T. The inset indicates a good linearity of the
temperature dependence of the linear resistivity. (b) Quasi-
2D VG scaling of the data measured at 0.1 T. The inset indi-
cates a good linearity of the temperature dependence of the
linear resistivity.
TABLE I: Quasi-2D VG scaling parameters at different fields.
µ0H (T) Tg(K) ν z
0.0 39.94 1.12 6.61
0.1 39.28 1.30 6.08
0.5 35.95 1.37 6.42
1.0 31.4 1.32 6.12
3.0 15.4 1.17 6.58
V. SUMMARY
We have measured I-V curves on high-quality MgB2
films at various magnetic fields and temperatures. At
magnetic fields below 5 T including the zero field, the
curves scaled well according to the quasi-2D VG theory
instead of the 3D model, in good agreement with the
multiband superconductivity of MgB2 contributed from
the strong 2D σ bands and weak 3D π bands. At the
fields above 5 T, the curves did not scale according to
any known VG scaling laws, accompanied by the disap-
pearance of a zero-resistance state. Based on our result
combined with recent tunneling experiments, a different
vortex state was suggested, namely, a state where the
vortices composed of the superfluid from the σ bands
move through the space filled with numerous quasiparti-
cles from π bands.
VI. ACKNOWLEDGMENTS
This work is supported by the National Science Foun-
dation of China, the Ministry of Science and Tech-
nology of China (973 project: 2006CB601000 and
2006CB921802), and the Knowledge Innovation Project
of the Chinese Academy of Sciences (ITSNEM). The
work at Penn State is supported by NSF under Grants
Nos. DMR-0306746 (X.X.X.), DMR-0405502 (Q.L.), and
DMR-0514592 (Z.K.L. and X.X.X.), and by ONR under
grant No. N00014-00-1-0294 (X.X.X.).
∗ Electronic address: hhwen@aphy.iphy.ac.cn
1 J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani,
and J. Akimitsu, Nature (London) 410, 63 (2001).
2 A. Rydh, U. Welp, A. E. Koshelev, W. K. Kwok, G. W.
Crabtree, R. Brusetti, L. Lyard, T. Klein, C. Marcenat, B.
Kang, K. H. Kim, K. H. P. Kim, H.S. Lee, and S. I. Lee,
Phys. Rev. B 70, 132503 (2004); A. E. Koshelev and A. A.
Golubov, Phys. Rev. Lett. 92, 107008 (2004); ibid, Phys.
Rev. B 68, 104503 (2003).
3 P. de la Mora, M. Castro, and G. Tavizon, J. Phys.: Con-
dens. Matter 17, 965 (2005); I. Pallecchi, M. Monni, C.
Ferdeghini, V. Ferrando, M. Putti, C. Tarantini, and E.
Galleani D’Agliano, Eur. Phys. J. B 52, 171 (2006).
4 Q. Li, B. T. Liu, Y. F. Hu, J. Chen, H. Gao, L. Shan, H.
H. Wen, A. V. Pogrebnyakov, J. M. Redwing, and X. X.
Xi, Phys. Rev. Lett. 96, 167003 (2006).
5 H. J. Choi, D. Roundy, H. Sun, M. L. Cohen, and S. G.
Louie, Nature (London) 418, 758 (2002).
6 M. Iavarone, G. Karapetrov, A. E. Koshelev, W. K. Kwok,
G. W. Crabtree, D. G. Hinks, W. N. Kang, E. M. Choi,
H. J. Kim, H. J. Kim, and S. I. Lee, Phys. Rev. Lett. 89,
187002 (2002).
7 K. H. P. Kim, W. N. Kang, M. S. Kim, C. U. Jung, H.
J. Kim, E. M. Choi, M. S. Park, and S. I. Lee, Physica C
370, 13 (2002).
8 S. K. Gupta, S. Sen, A. Singh, D. K. Aswal, J. V. Yakhmi,
E. M. Choi, H. J. Kim, K. H. P. Kim, S. Choi, H. S. Lee,
W. N. Kang, and S. I. Lee, Phys. Rev. B 66, 104525 (2002).
9 X. H. Zeng, A. V. Pogrebnyakov, A. Kotcharov, J. E.
Jones, X. X. Xi, E. M. Lysczek, J.M. Redwing, S. Y. Xu,
J. Lettieri, D. G. Schlom, W. Tian, X. Q. Pan, Z. K. Liu ,
Nature Mater. 1, 35 (2002).
10 H. H. Wen, S. L. Li, Z. W. Zhao, H. Jin, Y. M. Ni, W. N.
Kang, H. J. Kim, E. M. Choi, and S. I. Lee, Phys. Rev. B
64, 134505 (2001).
11 M. P. A. Fisher, Phys. Rev. Lett. 62, 1415 (1989); D. S.
Fisher, M. P. A. Fisher, and D. A. Huse, Phys. Rev. B 43,
130 (1991); D. A. Huse, M. P. A. Fisher, and D. S. Fisher,
Nature (London) 358, 553 (1992).
12 R. H. Koch, V. Foglietti, G. Koren, A. Gupta, and M. P.
A. Fisher, Phys. Rev. Lett. 63, 1511 (1989); R. H. Koch,
V. Foglietti, and M. P. A. Fisher, ibid. 64, 2586 (1990).
13 H. Yamasaki, K. Endo, S. Kosaka, M. Umeda, S. Yoshida,
and K. Kajimura, Phys. Rev. B 50, 12959 (1994).
14 V. L. Berezinskii, Sov. Phys. JETP 32, 493 (1970); J. M.
Kosterlitz, and D. J. Thouless, J. Phys. C 6, 1181 (1973).
15 H. H. Wen, P. Ziemann, H. A. Radovan, and S. L. Yan,
Europhys. Lett. 42, 319 (1998).
16 M. P. A. Fisher, T. A. Tokuyasu, and A. P. Young, Phys.
Rev. Lett. 66, 2931 (1991);
17 C. Dekker, P. J. M. Wöltgens, R. H. Koch, B. W. Hussey,
and A. Gupta, Phys. Rev. Lett. 69, 2717 (1992).
18 H. H. Wen, A. F. Th. Hoekstra, R. Griessen, S. L. Yan, L.
Fang, and M. S. Si, Phys. Rev. Lett. 79, 1559 (1997).
19 H. H. Wen, H. A. Radovan, F. M. Kamm, P. Ziemann, S.
L. Yan, L. Fang, and M. S. Si, Phys. Rev. Lett. 80, 3859
(1998).
20 Y. Z. Zhang, R. Deltour, J. F. de Marneffe, H. H. Wen, Y.
L. Qin, C. Dong, L. Li, and Z. X. Zhao, Phys. Rev. B 62,
11373 (2000).
21 H. Jin, H. H. Wen, H. P. Yang, Z. Y. Liu, Z. A. Ren, G. C.
Che, and Z. X. Zhao, App. Phys. Lett. 83, 2626 (2003).
22 D. R. Strachan, M. C. Sullivan, P. Fournier, S. P. Pai, T.
Venkatesan and C. J. Lobb, Phys. Rev. Lett. 87, 067007
(2001).
23 Y. Jia, H. Yang, Y. Huang, L. Shan, C. Ren, C. G. Zhuang,
Y. Cui, Q. Li, Z. K. Liu, X. X. Xi, and H. H. Wen,
arXiv:cond-mat/0703637 (unpubilished).
24 R. S. Gonnelli, D. Daghero, G. A. Ummarino, V. A.
Stepanov, J. Jun, S. M. Kazakov, and J. Karpinski, Phys.
Rev. Lett. 89, 247004 (2002).
25 M. R. Eskildsen, M. Kugler, S. Tanaka, J. Jun, S. M. Kaza-
kov, J. Karpinski, and Ø. Fischer, Phys. Rev. Lett. 89,
187003 (2002).
26 A. Anthore, H. Pothier, and D. Esteve, Phys. Rev. Lett.
90, 127001 (2003).
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