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PHYSICAL REVIEW B
VOLUME 57, NUMBER 6
1 FEBRUARY 1998-II
Out-of-plane Hall effect in YBa2Cu3O72d :
Vortex-glass behavior and scaling of c-axis and Hall resistivities
Yu. Eltsev* and Ö. Rapp
Department of Solid State Physics, Kungliga Tekniska Högskolan, SE-100 44 Stockholm, Sweden
~Received 18 November 1997!
The Hall resistivity, r zx , and longitudinal resistivity, r zz , have been studied in the mixed state of
YBa2Cu3O72d single crystals with transport current parallel to the c axis and magnetic field up to 12 T applied
along the CuO2 layers. Vortex-glass behavior is found for both r zx (T) and r zz (T) at small resistivities. A
scaling relation has been observed between r zx and r zz over a range in the B-T phase diagram that extends
beyond the vortex-glass critical regime. r zx (T) is negative at all temperatures around T c , demonstrating the
absence of a relation between scaling and a change of sign of the Hall resistivity. @S0163-1829~98!51606-2#
Since the discovery of high-T c superconductors ~HTSC!
the Hall effect in these compounds is a subject of intensive
experimental and theoretical studies. One of the most striking features of the Hall effect in HTSC is a sign change of
the in-plane Hall resistivity r xy near T c as a function of
temperature or magnetic field B, for B i c axis. This sign
change is from positive to negative with decreasing temperature in YBa2Cu3O72d ~YBCO!,1 while in the more anisotropic and weak pinning materials like Bi2Sr2CaCu2O8, 2
Tl2Ba2CaCu2O8, 3 and HgBa2CaCu2O6, 4 there is a second
sign change back to positive r xy at lower temperatures. Another remarkable aspect of the in-plane Hall effect attracting
great attention is a scaling relation between r xy and longitudinal resistivity r xx with r xy (T)5A @ r xx (T) # b . Luo et al.5
found b 51.760.2 in epitaxial YBCO films. Later similar
scaling has been observed also in single crystals of
YBa2Cu3O72d, 6 and Bi2Sr2CaCu2O8, 7 and in films of
Tl2Ba2CaCu2O8, 8 and HgBa2CaCu2O6, 4 with b in the range
1.5–2.
To explain these puzzling experimental results, several
different theoretical models have been proposed.9–11 Dorsey
and Fisher ~DF!9 suggested a scaling theory for the Hall
effect near the vortex-glass transition which was based on
the idea of ‘‘particle-hole’’ asymmetry. They showed that
the Hall resistivity should scale with a universal power of the
longitudinal resistivity. This relation contains a particle-hole
asymmetry exponent l v specially introduced in their theory.
A phenomenological model by Vinokur, Geshkenbein,
Feigel’man, and Blatter10 ~VGFB! suggests that scaling of
the Hall resistivity with an exponent b 52 is a general feature of any vortex state with disorder-dominated dynamics
~thermally assisted flux flow, vortex glass, etc.!. Wang,
Dong, and Ting11 ~WDT! developed a unified theory for the
flux motion in the mixed state of type-II superconductors in
the presence of pinning and thermal fluctuations. While DF
and WGFB consider the sign change of the Hall resistivity to
be a nonuniversal feature of vortex dynamics which in general should be considered in relation to a specific material,
WDT accounted for both scaling and sign reversal of the
Hall resistivity in their model. They further found b to decrease from 2.0 to 1.5 with increasing pinning strength.
0163-1829/98/57~6!/3237~4!/$15.00
57
Another interesting feature of the Hall effect in HTSC
which has not attracted so much attention is a negative sign
of the out-of-plane Hall response found in the normal state of
YBCO single crystals with B i ab in contrast to the positive
in-plane Hall effect (B i c) above T c . 12 Furthermore, above
T c the out-of-plane Hall resistivity r xz of YBCO is almost
temperature independent,12,13 in striking difference to the inplane Hall resistivity r xy ;1/T in the normal state.14 With
fields and currents both parallel to layers, Harris et al.15 also
found r xz to be negative below T c at all accessible temperatures in a magnetic field of 14 T.
We report on transport properties of YBCO with the current applied along the c axis, and B i ab. Vortex-glass scaling
was observed for both the c-axis resistivity r zz and the Hall
resistivity r zx . Furthermore, r zx was found to scale with
r zz , as r zx (T)5A @ r zz (T) # b over a range extending beyond
the vortex glass critical scaling. b was in the range 1.5–1.7
for two different samples, and A was found to be independent of magnetic field. r zx (T) is negative at all temperatures
near T c , indicating the absence of a relation between scaling
behavior and change of sign of the Hall resistivity.
Single crystals of YBa2Cu3O72d were grown by a selfflux method as described previously.16 Two twinned single
crystals were used. Sample c1, of dimensions 0.2930.21
30.32 mm3 with the largest size along the c axis, was naturally grown, and another one (c2) was cut parallel to c to
approximately the same size. The samples were oxygenated
in different ways: Sample c1 was annealed at 460, 400, 350,
and finally 300 °C in oxygen flow, for a few days at each
temperature. It exhibited a metalliclike behavior of resistivity
(d r c /dT.0). T c , defined as the midpoint of the resistive
transition was about 91 K with DT c ;0.2 K and
r c ~100 K!'4 m V cm. 17 Sample c2 was annealed at 450 °C
in air during 1 week. r c (T) of this crystal displayed a small
peak just above T c . This sample had a higher T c 591.6 K,
with DT c ;0.5 K and r c ~100 K!'12 m V cm. Contacts of
silver paint were made with current contacts covering both
ab-plane surfaces and three potential contacts in the form of
small dots on two opposite ac(bc) surfaces. The c-axis resistivity r c [ r zz was measured between two contacts aligned
along the direction of the transport current on one of the
ac(bc) surfaces. The out-of-plane Hall voltage was meaR3237
© 1998 The American Physical Society
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YU. ELTSEV AND Ö. RAPP
FIG. 1. Out-of-plane longitudinal r zz ~top panel! and Hall r zx
~bottom panel! resistivities of sample c1 measured simultaneously
as a function of temperature in magnetic fields up to 12 T. Inset in
top panel: Arrhenius plots of r zz (T) for the same magnetic fields.
The line is a linear fit to data at B512 T. Inset in bottom panel:
In-plane Hall resistivity r xy vs temperature for the same sample c1.
The contacts used for out-of-plane measurements were carefully
mechanically removed, and replaced by new ones in the usual inplane geometry. Magnetic fields up to 6 T as indicated were applied
perpendicularly to layers.
sured across two contacts on the opposite ac(bc) surfaces,
and r zx was obtained from the antisymmetric part of the
voltage response under magnetic field reversal. A dc picovoltmeter was used to measure both longitudinal and transverse voltages. The current density j>25 A/cm2 was well
inside the range of linear response for both voltages.
Shown in Fig. 1 are the temperature dependences of the
c-axis resistivity r zz ~top panel! and the Hall resistivity r zx
~bottom panel! of sample c1 measured in magnetic fields up
to 12 T. Data for sample c2 are similar, except for a slightly
increased width of the superconducting transition and higher
values of both resistivities. r zz displays a pronounced broadening of the superconducting transition in a magnetic field,
similar to the usually observed behavior of the in-plane longitudinal resistivity r xx , while r zx is rather different from
the in-plane Hall resistivity r xy , measured on the same crystal, and shown in the inset in the bottom panel of Fig. 1. At
a fixed value of B and with increasing temperature from
below T c , r zx (T) displays a similar negative anomaly as
r xy (T). Above T c , however, r zx saturates at a small negative value, in contrast to r xy which exhibits the well-known
sign reversal. The linear increase of r zx with B above T c ,
and a strongly nonlinear behavior at lower temperatures are
57
FIG. 2. Main panel: (d log10rzz /dT) 21 vs T for sample c2 at
B512 T. A fit to the vortex-glass model with T g 584.4 K and
@v (z21) # 21 50.15 is shown by the line. T * marks the onset of
deviations for increasing temperatures. Upper inset: Log-log plot of
r zz (T) ~filled circles! and u r zx (T) u ~open circles! vs (T2T g )/T g for
B512 T with T g 584.4 K from the main panel. Lines are fits to the
vortex-glass model with exponents as indicated. Arrows correspond
to T * in the main panel. Lower inset: critical exponents g and
n (z21) for different fields.
similar to the results for r xz of Ref. 15. Furthermore, in
accordance with the Onsager relation, our r zx (B,T) data for
sample c1 are in a good quantitative agreement with r xz data
reported by Harris et al.15 for their YBCO single crystals
with approximately the same oxygen content.
We start to analyze the results with the behavior of r zz (T)
near T c in a magnetic field applied along the CuO2 planes
which to our knowledge has not been discussed in detail
before. In the inset of the top panel of Fig. 1, Arrhenius plots
of r zz as a function of temperature are shown. In the middle
part of the transition there is a region where ln(rzz) varies
linearly with 1/T, which thus reflects activatedlike behavior,
r zz (T);exp(2U/kBT). On the other hand, with the further
decrease of temperature the slopes of the curves of lnrzz vs
1/T rapidly increase, suggesting that there is no simple thermally activated behavior when r zz →0.
According to the vortex-glass model,18 the resistivity vanishes as r ;(T-T g ) n (z21) when temperature approaches the
glass temperature T g . Therefore the inverse logarithmic derivative of the resistivity (dlog10rzz /dT)21 vs temperature
should be a straight line which extrapolates to zero at T g
with a slope 1/n (z21). In the main panel of Fig. 2 we
present data for sample c2 at B512 T consistent with a
vortex-glass model with T g 584.460.2 K and 1/n (z21)
50.15. A power-law dependence r zz ;(T-T g ) n (z21) is observed in a narrow temperature range below T * inside the
critical scaling region: DT5T * 2T g '2.5 K. This region
of glassy scaling diminishes with decreasing magnetic field
~not shown!, which also is in agreement with the vortexglass model.18
The top inset in Fig. 2 illustrates that the proper T g has
been found, by showing the low resistivity part of the data
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OUT-OF-PLANE HALL EFFECT IN YBa2Cu3O72d : . . .
FIG. 3. Log10u r zx u vs log10r zz for sample c1 ~filled symbols!
and sample c2 ~open symbols! obtained at varying temperatures for
B54, 8, and 12 T. Lines are fits for both samples. r zz (T sc ) marks
the onset of deviations from u r zx (T) u ; @ r zz (T) # b for sample c2 at
12 T.
from the main panel in the form of log10rzz(T) vs
log10(T/Tg21) as a straight line with a slope of n (z21)
56.6. Data for other fields are similar, and the corresponding
critical exponents are shown in the lower inset in Fig. 2 to be
field independent within error bars. The scaling exponent
n (z21)56.561.0 obtained here is in remarkable agreement
with the value of n (z21)56.5 found previously from analysis of the vortex-glass behavior of the in-plane longitudinal
resistivity of YBCO single crystals19 and films5,20 with B i c,
in contrast to a much smaller value of the same exponent of
about 1.4 extracted from measurements of r xx (T) with both
B and j in the planes.21 This striking difference may reflect
the importance for the out-of-plane motion of vortices of the
intrinsic pinning due to the layered structure of YBCO. With
B i ab and j i ab the Lorentz force exerted on the vortices by
transport current pushes them perpendicularly to ab planes
through a periodic pinning potential, while with B i ab and
j i c intrinsic pinning seems to be negligible since the Lorentz
force is parallel to layers.
In the upper inset in Fig. 2 we also show log10urzx(T)u vs
log10 (T2Tg)/Tg at B512 T using the same value of T g
584.4 K as determined from the analysis of r zz (T). Similarly to the c-axis resistivity, the out-of-plane Hall resistivity
exhibits a power-law dependence of T2T g in a restricted
temperature range above T g ; u r zx (T) u ;(T2T g ) g . Values of
g obtained at a few different fields are shown in the lower
inset in Fig. 2. Errors increase with decreasing magnetic field
due to contraction of the critical scaling region. g is within
9.961.0 at all fields, in fair agreement with the corresponding exponent for u r xy u of 10.461.5 obtained at 3.7 T for
B i c. 5 Thus, also for the transport current along the c axis
and with B i ab, the c-axis and Hall resistivities are consistent with the vortex-glass model.9,18
The power-law behavior of r zz and r zx as a function of
T2T g suggests a scaling dependence between them:
r zx (T)5A @ r zz (T) # b . In Fig. 3 we plot log10urzx(T)u vs
log10rzz(T) for a few magnetic fields. For each sample, data
obtained at different B collapse onto a single straight line.
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We thus obtain field-independent values of b 51.6560.1 for
sample c1 and b 51.560.1 for sample c2. These results are
close to those previously obtained from the in-plane Hall
effect, with b 51.760.2 for YBCO films,5 b 51.560.1 for
YBCO single crystal with columnar defects,22 and below b
52.060.2 reported for unirradiated YBCO single crystal
with a lower level of defects.22 From the data for sample c2
at B512 T shown in Fig. 3 and the upper inset in Fig. 2 one
can see that the scaling relation between r zx (T) and r zz (T)
sets in the region below T * when a finite r zx (T) can be
observed and with further increase of temperature extends
beyond the region of critical scaling up to r zx (T sc ) and
r zz (T sc ), exceeding by nearly one order of magnitude the
corresponding resistivity values at T * .
The observed vortex-glass critical scaling of both r zz (T)
and r zx (T) near T g and the scaling dependence between
them are consistent with DF theory9 except for the fact that
our relation r zx ; r bxx extends far above the region of glassy
scaling, to which scaling between r zz and r zx should be restricted according to DF. An extended range of scaling dependence between r zz and r zx seems to be in agreement with
the VGFB ~Ref. 10! model where the scaling law is universal, independent of the specific vortex structure, whether a
vortex liquid or a vortex glass. On the other hand b 52, as
strictly obtained in this theory, exceeds b 51.50– 1.65 as
found by us. WDT ~Ref. 11! theory gives an exponent which
decreases with increasing disorder and reaches b 51.5 for
strong pinning. This is consistent with our results, and is
qualitatively supported by the decrease of b from 1.65 for
sample c1 to 1.5 for sample c2, where a larger disorder is
suggested by the higher resistivity. Furthermore, from Fig. 3
it can be deduced that the constant A in r zx (T)
5A @ r zz (T) # b is smaller for the high resistivity sample,
which is also in qualitative agreement with WDT. In contrast
to their model, however, the absence of a sign reversal of
r zx (T) rules out a relation between change of sign and scaling of the Hall resistivity. It can also be noted that the coefficient A is field independent for each sample. Theoretically
this question is controversial, with arguments advanced both
in favor of10 and against11 a field-independent A.
The rather good agreement between our results and the
conclusions of theoretical models of the Hall effect in the
mixed state of HTSC ~Refs. 9–11! seems to be surprising.
All these models were developed for the case of quenched
disorder while in our experimental geometry with B i ab
planes in addition to the usual randomly distributed pinning
sites ~e.g., oxygen vacancies!, the layered structure of YBCO
parallel to the magnetic field provides for an intrinsic pinning
potential periodic along the c axis. The significance of the
intrinsic pinning for the out-of-plane motion of vortices is
well illustrated by the considerable difference between the
values of the scaling exponent determined from the analysis
of the power-law dependence of longitudinal resistivity of
YBCO with j i ab in Ref. 21 and j i c in this study ~1.4 and
6.5, respectively!. On the other hand, from measurements of
the angular dependence of the Hall effect of YBCO single
crystals, Harris et al.23 have found strong experimental support for the continuous anisotropic mass model,24 which considers intrinsic pinning to be negligible.
Another important issue related to our experimental geometry with B i ab planes is the nature of vortices directed
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YU. ELTSEV AND Ö. RAPP
57
along the superconducting layers of YBCO. Although nearly
fully oxygenated YBCO with g 5(m c /m ab ) 1/2;7 is known
to be much less anisotropic than the other HTSC ~e.g., g
550– 200 for Bi2Sr2CaCu2O8!,25 the coherence length j c
;2 Å ~Ref. 26! is small in comparison with the distance
between the CuO2 layers d58.4 Å, which opens a possibility of Josephson coupling between superconducting CuO2
layers and Josephson-type rather than Abrikosov-type vortices parallel to ab planes. However, in a study of the intrinsic
Josephson effect in different HTSC, Kleiner and Muller27 did
not find any signs of Josephson effects in YBCO in contrast
to the more anisotropic Bi2Sr2CaCu2O8 and Tl2Ba2CaCu2O8
compounds which probably favors a usual Abrikosov-like
nature of vortices directed parallel to CuO2 planes of YBCO.
This discussion clearly demonstrates the need for more
work on the out-of-plane Hall effect in the mixed state of
HTSC including detailed theoretical study of this effect as
well as further experiments, e.g., in samples with different
anisotropies.
In conclusion, we have measured longitudinal and Hall
resistivities of YBCO single crystals in magnetic fields parallel to the layers up to 12 T and with current parallel to c.
At low resistivities r zz (T) and r zx (T) are both consistent
with the vortex-glass model.9,18 r zx (T) and r zz were found to
scale as r zx (T)5A @ r xx (T) # b with b 51.50–1.65 and a fieldindependent A, over a region extending beyond the vortexglass scaling. In contrast to r xy , r zx is negative at all temperatures around and above T c . This result gives a clear
experimental demonstration that scaling behavior and sign
change of the Hall resistivity are of different origins.
*Also at the Solid State Physics Department, P. N. Lebedev Physi-
15
cal Institute, 117924, Moscow, Russia.
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This work has been supported by the Natural Science Research Council, by the Göran Gustafsson Foundation, and by
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