VOLUME 78, NUMBER 16
PHYSICAL REVIEW LETTERS
21 APRIL 1997
Superfast Vortex Creep in YBa2 Cu3 O72d Crystals with Columnar Defects:
Evidence for Variable-Range Vortex Hopping
J. R. Thompson,1 L. Krusin-Elbaum,2 L. Civale,3 G. Blatter,4 and C. Feild2
1
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
and Department of Physics, University of Tennessee, Knoxville, Tennessee 37996
2
IBM Research, Yorktown Heights, New York 10598
3
Centro Atomico Bariloche –CNEA, 8400 Bariloche, Argentina
4
ETH-Hönggerberg, CH-8093 Zürich, Switzerland
(Received 17 June 1996)
We observe a large peak in the vortex creep rate of YBa2 Cu3 O72d crystals with columnar defects.
The peak appears at the crossover from the regime where vortices are well localized on columnar pins
to a regime where vortex-vortex interactions dominate, and it is most prominent (nearly 12%) at low
fields, near T , 40 K. Near this crossover, the current dependent creep activation energy UsJd shows
distinct and unusual behavior (“pinch”). On the high-J side of the pinch, the glassy exponent m , 1y3
and large creep rate are characteristic of variable range vortex hopping.
[S0031-9007(97)02959-1]
PACS numbers: 74.60.Ge, 74.72.Bk
A limit on technological advances of high temperature
superconductors comes from the intrinsically low critical
current densities Jc and from the ease with which magnetic vortices can move, i.e., from high (“giant”) vortex
creep rates [1]. A most efficient way to pin vortices is with
columnar defects [2–5] created via swift particle irradiation. A major surprise, however, is the thermal motion of
vortices in the presence of columnar tracks [6–9]; it is not
much reduced [9], it is regime specific, and can even be
enhanced for certain configurations of the tracks [8].
Some guidance to this complex vortex dynamics comes
from the work of Nelson and Vinokur [10]. By analogy
to a problem of localization of 2D bosons, they arrive at a
“Bose-glass” phase diagram from which distinct dynamic
regimes evolve via nucleation and motion of vortex kinks
[10]. Several features of the Bose-glass H-T diagram were
confirmed recently in heavy-ion-irradiated YBa2 Cu3 O72d
(YBCO) crystals [11–13]. In particular, we now have experimental evidence for the “accommodation field” B? sT d
[12], which separates a regime of strong localization of
single vortices and a collective regime where vortex-vortex
interactions prevail [10]. We also know that the columnar
pins have less than “ideal” pinning efficiency [12], and that
there is a significant entropic smearing of the pinning potential [1], allowing easy depinning at a fairly low s,40 Kd
temperature [12].
The central point of this paper is the origin of a striking dynamical feature in YBCO with columnar defects,
namely, a large peak in the current decay rate vs temperature [9,12] at the crossover into the collective pinning
regime at B? sT d. This peak is most pronounced at low
fields, where the maximum vortex creep rate is 6 times
greater than in the virgin samples which exhibit a welldocumented “plateau” [14] in the same temperature range.
Here we show that the creep peak is a consequence of two
separate (albeit related) processes. One is a superfast relaxation near the depinning temperature helped by a signifi0031-9007y97y78(16)y3181(4)$10.00
cant dispersion in the track pinning energies [10,15]. The
distinct current dependence of the activation energy UsJd
indicates that this creep proceeds via variable-range vortex
hopping (VRH) [10], in analogy to variable-range conduction in doped semiconductors [16]. The VRH process is
cut off by the crossover into the collective regime, where
the creep of vortex bundles becomes more sluggish [1,17].
Measurements were performed on two well characterized YBCO crystals of ,1 mm size and ,20 mm thick
along the c axis, irradiated at the TASCC facility (Chalk
River, Canada) to have field-equivalent defect densities
(matching fields) BF ­ 2.4 and 4.7 T [11,12]. The field,
temperature, and time dependence of the persistent current
density JsH, T , td was obtained from the irreversible
magnetization MsH, T , td [12] using the critical state
model [18]. MsH, T, td was measured with a SQUID
magnetometer in fields up to 6.5 T applied along the
direction of the incident beam.
Figure 1 shows JsT , td for the YBCO crystal with BF ­
2.4 T, for three values of magnetic field. At each temperature, the crystals were prepared (after zero-field cooling) in
a fully developed critical state by first increasing the field
to the maximum of 6.5 T and then reducing it to the target value. The time evolution of JsH, Td was recorded
during times 60 , t , 7200 sec from the paramagnetic
branch of the hysteresis loop [17]. The collection of data
points in this time interval at each temperature forms vertical traces shown in the figure—the length of each trace
indicates the decay in J. At low fields this decay is nonmonotonic in temperature, as seen in Figs. 1(a) and 1(b),
where the segments are longest at intermediate temperatures. This is reflected in the normalized thermal creep rate
S ­ 2d ln Jyd ln t (same figure), which has a large peak
at ,40 K at low fields. At m0 H ­ 0.5 T the maximum
creep rate is nearly 12%, well above the 2% measured at
higher temperatures (and above the plateau values in unirradiated YBCO crystals [14]). As field increases, the creep
© 1997 The American Physical Society
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PHYSICAL REVIEW LETTERS
FIG. 1. Persistent current density JsT , td and the corresponding relaxation rate SsT d for a YBCO crystal irradiated with
1 GeV Au (see Ref. [12]) to a dose of 2.4 T in fields (a) 0.5 T,
( b) 1.5 T, and (c) 4 T parallel to defects. Both normalized and
bare relaxation rates are fastest well below ,0.5BF near 40 K.
Inset: MsHd at 5 K. Arrows indicate the field-sweeping protocol to assure the full critical state (see text).
peak shifts to lower temperatures and disappears entirely
above the matching field BF . The peak position traces
the “accommodation” field B? sT d shown in Fig. 2, which
bounds the regime where vortices are pinned individually
[12]. B? sTd can also be obtained [12] from the maximum
change of JsT d and the onset of the 1yH field dependence
[1] of JsHd. At low temperatures B? sT d , BF , decreasing with T and becoming almost zero [19] at the depinning
temperature Tdp , 40 K; beyond Tdp the pinning is collective [1,12]. The time decay of the persistent current of
Fig. 1(a) is visualized in Fig. 2 along the lns1yJd axis—
the decay is largest near Tdp . The puzzling question is
why the creep rate at low fields is so high at temperatures
well below Tc and why there is a peak.
To address this we use Maley’s scheme [20] to determine the current dependence of the effective activation
energy UsJd. This scheme allows an analysis of the relaxation data without a priori assumptions for the current
and field dependence of U. Briefly, the procedure uses the
master rate equation UsJ, T d ­ 2kT flnsdMydtd 2 Cg.
Here C ­ lnsBvaypLd is a temperature independent
constant [20] uniquely fixed by requiring that, at low temperatures, U is a continuous function of J; B is a mag3182
21 APRIL 1997
FIG. 2. 3D diagram of pinning and creep in YBCO with
BF ­ 2.4 T. Accommodation field B? sT d is from the experiment in Ref. [12]. The regimes of single-vortex (svp) and
collective (cp) pinning are indicated in the B-T plane. Relevant
creep regimes are constructed along the lns1yJd (time) axis
from the experimental values of crossover currents (see text).
Theoretical m values are also indicated in the collective creep
(cc) and in the “dressed” vortex [1] regimes. The time decay of
JsT d at m0 H ­ 0.5 T is largest near Tdp . It starts out at low
T in the half-loop regime sm , 1d, passes the variable-rangehopping regime (VRH) sm , 1y3d, and exits on the collective
creep side with sm , 1.5d.
netic induction, v ­ 2pyt0 is the microscopic attempt
frequency [20], a is the hop distance, and L is the sample
dimension. [The value of C ­ 26 allows an estimate
of the microscopic hopping time [1] t0 ; e.g., at m0 H ­
0.5 T, with a , 100 Å [20], and L , 0.1 cm, we obtain
t0 , 3 3 10211 sec.] In order to maintain “piecewise”
continuity at high T, we divide UsJd by a thermal factor gsTd # 1 (inset of Fig. 3), containing the temperature dependence of the superconducting parameters [21].
Plots of UsJ, Td for several values of magnetic field are
shown in Fig. 3. A “pinch” in UsJd, indicating a dynamic
crossover, is evident for both crystals. This pinch is a reflection of the anomalous creep peak near B? sT d [12]; it
occurs at UsJd , 1000 K and only below BF .
So far we have used the Maley analysis only as a convenient, model-independent, and compact way to present
the relaxation data. Now we explore the connection with
the glassy picture of vortex transport [1,10]. In this picture, U is related to the instantaneous current density J as
UsJ, T d ­ U 0 sTd fsJc yJdm 2 1g, where U 0 is the characteristic pinning energy [1]. The value of the glassy exponent m is regime specific and characterizes the creep
process. This expression with m ­ 1 fits well our experimental curves of UsJd at high currents (low T), as
shown in Fig. 4 for m0 H ­ 0.5 T. It departs from the
data at ,23 K, corresponding to J , 4.6 3 106 Aycm2 .
Above this temperature, J ø Jc , so ln UsJd . ln U 0 1
msln Jc 2 ln Jd and thus D ln UyD ln J ­ m. It is clear
VOLUME 78, NUMBER 16
PHYSICAL REVIEW LETTERS
21 APRIL 1997
FIG. 4. UsJd for the BF ­ 2.4 T crystal in a 0.5 T magnetic
field. The solid line is the fit to the full glassy expression for
UsJd (see text) with m . 1. The slope m , 1y3 (dashed line)
fits the data well between 23 and 40 K. Crossover currents are
indicated by the arrows. Inset: Fit to variable-range hopping
[Eq. (3)] with m ­ 1y3 (see text) is shown as the solid line.
The decreased rate on the high-T side of the peak is due to
slower creep in the collective regime.
FIG. 3. Effective activation energy UsJd for two YBCO
crystals with BF ­ 2.4 T (top), and 4.7 T (bottom). The
pinch at the crossover into the collective regime at B? is
evident at U , 1000 K for both crystals. Top inset: msHd
for both crystals. The behavior is nearly identical for the
same values of the fiducial field ByBF . At low fields, the
value of glassy exponent m , 1y3 on the right (low-T , high-J)
side of the pinch. msHd increases smoothly on crossing the
accommodation field B? to above 1 in the collective regime
(above BF ). Bottom inset: thermal factor gsTd (see text) is the
same and structureless below and above B? .
from Figs. 3 and 4 that large segments of ln UsJd vs
ln J are linear, allowing the estimate of m [22]. At
m0 H ­ 0.5 T, above the pinch (for J between ,s1 4d 3
106 Aycm2 ), the slope m . 1y3. Below the pinch sT .
40 Kd, m . 1 is much larger. The crossover into the collective regime is illustrated in the top inset of Fig. 3. At
low fields (below 0.5BF ), msHd is close to 1y3 and is
weakly field dependent. A smooth increase of m across
the boundary at B? reflects the increasing effect of vortexvortex interactions. At high fields, m . 1 settles until the
irreversibility line is reached [indicated by the curvature in
UsJd at the lowest currents].
Generally, the vortex creep process can be viewed as
a sequence of (i) nucleation of half-loop excitations and
(ii) the subsequent expansion of half-loops with decaying
current [1]. In the Bose-glass scenario [10], the half-loop
expansion is characterized by a glassy exponent m ­ 1.
At sufficiently low fields and below Tdp , this expansion will eventually transfer the vortex segment from one
columnar pin to another. Further relaxation will spread
the resulting pairs of vortex kinks apart. If the pins are
not identical [2], nearest-neighbor hopping may not be favorable; the transfer may be to a lower-energy defect farther away. The dispersion in track energies (undoubtedly
present in a real sample [12,15]) will give rise to variablerange hopping of the vortex lines [10], which is characterized by m ­ 1y3 [10]. In the collective regime (above
Tdp ) the creep should be more sluggish and m is larger, as
is seen in Fig. 2. Thus, from our data we conclude that,
for example, at m0 H ­ 0.5 T, the crossover from the halfloop (hl) regime to the VRH regime occurs at a current
density Jhl at which the value of m crosses from 1 to 1y3
(see Fig. 4), and the crossover from VRH to the collective
regime occurs at a lower current JVRH near the pinch.
To test the above picture, we estimate from theory [1,10]
first the crossover currents and then the relevant energy
scale. We focus on the sample with BF ­ 2.4 T, and then
for consistency we check the estimates for the larger pin
density. The crossover current Jhl from the half-loop to
the VRH regime is given by [1]
j g
Jhl
­
,
Jc
d ´r
(1)
p
where j is the coherence length, d ­ F0 yBF is the
mean defect spacing, ´r is the track pinning energy, and
g measures the dispersion of pinning energies [10]. We
take Jc ­ s´r y´0 dJ0 , where ´0 ­ sF0 y4pld2 is the line
energy and J0 is the depairing current density [1]. The
basis for the estimate will be “nonideal” pinning efficiency h . 0.2 , 1 [11,12], so that the low temperature
Jc ­ 0.2J0 . 6 3 107 Aycm2 and ´r ­ h´0 . Assuming gy´r , 1 [10], with j , 20 Å and d ­ 300 Å, we
obtain Jhl yJc . 1y15 or Jhl . 4 3 106 Aycm2 , remarkably close to the experiment (see Fig. 4). The crossover
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PHYSICAL REVIEW LETTERS
current from VRH into the collective regime is [1]
µ
∂ µ
∂
JVRH
B 3y4 ´r 3y4
­
,
(2)
Jhl
BF
4hg
and thus JVRH . Jhl y4 . 1 3 106 Aycm2 , again in good
agreement with the experiment. The energy scale associated with this crossover is the double-kink generap
p
tion energy [1,10] UVRH ­ Edk ­ ed ´0 ´r ­ e´0 d h.
For YBCO, with the anisotropy parameter e , 1y5 and
ls0d . 1400 Å [1], we estimate Edk , 3000 K. This
may be an overestimate, since creep will involve not the
average spacing d, but the smallest possible pin distance.
Thus, the energy scale is also in reasonable agreement with
the UsJd , 800 1000 K value near the crossover.
Finally, we compare the samples with different pin densities. The data for BF ­ 4.7 T show a crossover at nearly
the same UsJd and a much smaller current JVRH , 3 3
p
105 Aycm2 . A simple scaling UVRH ~ d h s´r ygd ~
dygh 3y2 and JVRH ~ gydJ0 [1] does not account for a
factor of 4 downshift in current if the dispersion is independent of BF . To fit the data consistently, one needs
larger dispersion gy´r , 3, plausibly due to more numerous track overlaps at larger BF [12].
Thus, we propose the following picture for the huge relaxation peak near the depinning temperature in a superconductor with columnar defects. Near Tdp the entropic
smearing of the pinning potential is large. Hence, there
is prolific generation of double kinks, which will hop to
optimal pinning sites that can be closer (or more distant)
than d, due to dispersion in ´r [15]. This will result in a
large dissipation on the approach (on warming) to Tdp , as
indicated by the normalized creep rate [1],
kT
.
(3)
0
UVRH
1 mkT lnstyteff d
p
0
Here UVRH
sT d ~ ´0 hfsTd [1,12] is reduced by the
entropic smearing factor [1] fsT d ~ B? sT dyBF [12], and
thus must vanish at Tdp (see Fig. 2). The creep rate
should saturate. From Eq. (3), with m . 1y3 and (with
teff > 103 t0 [23,24]) lnstyteff d . 26, the saturation will
occur at S , 0.1, close to the observed maximum (inset
of Fig. 4). The saturation is disrupted, however, by a
crossover into the collective regime, where m . 1.5 and
the creep is much slower with S ø 0.02. In summary,
a VRH mechanism is supported by the interpolation
formula with the glassy exponent independently obtained
from distinct UsJd and by the relevant current and energy
scales in the context of the entire H-T diagram for
samples with different pin densities.
There still remains a puzzling issue as to why, at low
T, B? sTd is decreasing much faster (linearly with increasing T [12]) than suggested in the theory [10]. One intriguing possibility is a d-wave character of the superconducting order parameter. The pinning energy ´r sT d ~
fsTdyl2 sT dj 2 sT d [1,10]. And if both lsT d and jsT d increase much faster with T than for a BCS-like superconductor, the pinning will be reduced faster. With lsTd from
S­
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21 APRIL 1997
Hardy et al. [25], we reproduce the linear slope of B? sT d
below Tdp [12] and may account for easier depinning.
L. K.-E. acknowledges useful discussions with D. R.
Nelson and V. M. Vinokur, and the expert assistance of
D. Lopez. Work at ORNL was sponsored by the DMS,
U.S. DOE under Contract No. DE-AC05-96OR22464
with Lockheed Martin Energy Research Corp. We thank
J. Hardy and J. Forster at TASCC (Chalk River, Canada)
for their help and the provision of irradiation facilities.
TASCC is supported by AECL Research.
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[22] Note that UsJd obtained from nonthermally cycled data
has a “real” meaning only if there is no crossover to a
different pinning regime. However, m depends only on
the slopes of the segments and is well defined.
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[24] The characteristic time teff which enters the interpolation
formula is a macroscopic quantity [1]. For a slab of
thickness d we have teff ø sTd 2 ycj≠Uy≠JjaHdt0 . With
≠Uy≠J , U0 yJc and taking TyU0 , 0.02, dya , 105 ,
and H , H ? , Jc dyc, we estimate teff , 103 t0 . This
estimate is not critical, since teff enters in the logarithm.
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