PHYSICAL REVIEW B
VOLUME 57, NUMBER 17
1 MAY 1998-I
Activation energy for vortex motion in anisotropic superconductors
M. F. Laguna, E. A. Jagla, and C. A. Balseiro
Comisión Nacional de Energı́a Atómica, Centro Atómico Bariloche and Instituto Balseiro, 8400 San Carlos de Bariloche, Argentina
~Received 24 October 1997!
We study the transport properties of highly anisotropic superconductors with different kinds of topological
disorder using Josephson-junction networks. The model describes a stack of weakly coupled superconducting
planes in the presence of an external magnetic field perpendicular to them. At low temperatures the resistivities
for current flowing perpendicularly and parallel to the external field show a thermally activated behavior. For
systems with point defects, the activation energy is the same for the two components of the resistivity, while
in systems with columnar defects they may be different. The results are interpreted in terms of a simple model
and compared with experimental results. @S0163-1829~98!05913-X#
I. INTRODUCTION
High-temperature superconductors are layered materials
in which charge dynamics is mostly confined to CuO planes.
This intrinsic anisotropy strongly influences the structure of
vortices in the mixed state.1
The simplest model for vortices in these systems when the
external magnetic field is applied along the c axis of the
sample considers pancake vortices in each plane.2,3 The
model includes intra- and interplane interactions of the pancake vortices and assumes they are electromagnetic in nature. In the absence of intrinsic disorder, at low temperatures
the interplane interactions generate a stack of pancakes that
can be viewed as an Abrikosov vortex. On the other hand,
the intraplane interactions generate an ordered state: the
Abrikosov lattice. As the temperature increases, thermal
fluctuations can destroy both the in-plane and the interplane
long-range order. In the high-temperature phase, the picture
of Abrikosov vortices threading the sample in the direction
of the field breaks down: thermal fluctuations destroy the
stacking long-range order and the vortex is cut in small segments or in individual pancakes.
The pancake model neglects the Josephson coupling between neighboring planes that can change the scenario in a
rather dramatic way.4 Josephson coupling contributes to generate the stacking order of the pancake vortices. However
when thermal fluctuations or point defects produce misalignment of the stack, tunneling currents flow between planes to
form a short Josephson vortex connecting the pancakes. In
this way, even in the absence of stacking order, vortex lines
can be defined. If these lines are disentangled, there is a
unique connection between two pancakes, and vortex lines
are well defined, are continuous and cannot be broken in
small pieces. If so, the loss of superconducting coherence is
not given by the lack of long-range order in the position of
pancakes along the c axis. Instead the superconducting coherence along the field direction is destroyed by the entanglement of vortex lines and the thermal excitation of vortex loops.5–7 In previous works we have shown that, for
temperatures larger than a characteristic temperature T p ,
these excitations produce vortex lines that percolate in the
sample perpendicularly to the external field.8 When an external current flows in the direction of the field, it produces a
0163-1829/98/57~17!/10884~5!/$15.00
57
Lorentz force on these percolating paths and their motion
generates dissipation, a signature of the lack of superconducting coherence. The coherent length in the c direction is
then given by the mean distance between percolating vortex
lines, as the temperature increases above T p these lines proliferate in the sample and the coherence length decreases. At
high enough temperatures, when the coherence length is of
the order of the interplane distance, both models describe
essentially the same physics and the picture of pancake vortices becomes an appropriate and simple scenario.
Experimentally low-temperature tails in the temperature
dependence of the resistivity for magnetic fields perpendicular to the planes are observed in the highly anisotropic
BiSrCuO samples,9,10 which are due to the thermally activated motion of vortices. It has been shown that in single
crystals the thermally activated behavior of both the in-plane
( r ab ) and out-of-plane ( r c ) resistivities is characterized by
the same value of the activation energy. This behavior contrasts with that of crystals with columnar defects where, for
fields smaller than the matching field B f , the activation energy for the out-of-plane resistivity r c is larger than that of
the in-plane resistivity r ab . 10
Recently Koshelev11 studied the behavior of the resistivities r ab and r c starting with a model of decoupled pancake
vortices and including the intraplane Josephson coupling perturbatively. He showed that, in systems with point defects,
the activation energy is the same for the two components of
the resistivity. In this work we present numerical simulations
for the transport properties when the external current flows
parallel and perpendicularly to the external magnetic field in
highly anisotropic systems with different kinds of topological disorder.
II. MODELING TRANSPORT PROPERTIES
We performed numerical simulations on the threedimensional ~3D! Josephson-junction array ~JJA! model on a
stacked triangular network. Each junction between nearestneighbor nodes is modeled by an ideal junction with critical
current I c shunted by a normal resistance R. The dynamics
of the model consists of the time evolution of the phase of
the order parameter w i (t) defined at each node, and is given
by the following Langevin equations:12
10 884
© 1998 The American Physical Society
ACTIVATION ENERGY FOR VORTEX MOTION IN . . .
57
1 ] ~ D w ii 8 !
1 h ii 8 ~ t ! ,
R
]t
~1!
j ii 8 ,
(
$i %
~2!
j ii 8 5I c sin~ D w ii 8 ! 1
j iext5
8
here j ii 8 is the current between nodes i and i 8 , D w ii 8 [ w i
2 w i 8 2A ii 8 is the gauge invariant phase difference, A ii 8 is
the vector potential of the external magnetic field, and
h ii 8 (t) is an uncorrelated Gaussian noise which accounts for
the thermal fluctuations. Equation ~2! guarantees the current
conservation at each node ~the sum is over the neighbor
nodes i 8 of the node i) with j iext the external current applied
at the node. Equations ~1! and ~2! are integrated using a
second-order Runge-Kutta algorithm. We take periodic
boundary conditions in the directions perpendicular to the
external current, and open ~pseudoperiodic!8,13 boundary
conditions for the a ~c! direction when external current is
applied in the a ~c! direction.
A vortex in a given plaquette is characterized by a circulation of 2 p of the superconducting phase around this
plaquette. In this model the only source of dissipation is
vortex motion. When a vortex passes between two contacts,
the phase difference at the contacts increases by 2 p and,
according to the following Josephson equation, generates a
voltage V:
V5
K
L
]~ w 12 w 2 !
,
]t
~3!
where w 1 and w 2 are the phases at the contacts and ^ ••• &
indicates time averaging. We calculate the resistivity by injecting a small external current and calculating the voltage.
The external current is chosen to have a small value ~usually
1/100 of the critical current of the junctions! in order to
detect only the linear resistivity of the system.
Anisotropy is introduced by reducing the critical current
of the junctions along the c axis by a factor h 2 , and increasing their normal resistance by the same factor. Disorder is
simulated by randomly varying the critical current of the
junctions through the lattice. As the energy of the vortices
decreases when they are located in a region of low critical
currents, the random distribution of critical currents provides
a nearly random pinning potential for vortices. The critical
current distribution is taken flat between I max
and I min
c
c , the
maximum and minimum value of the critical current through
the sample. We characterize the disorder by a parameter
min
max
min
D[(I max
c 2Ic )/(Ic 1Ic ).
The 3D JJA has been used to study the first-order melting
of the vortex lattice in clean systems.14 In addition the effect
of disorder on the melting transition was analyzed in some
detail using both thermodynamical and transport properties
to characterize the different phases.15
The anisotropy-disorder phase diagram shows three types
of behaviors depending on the parameters:
~i! For moderate or high anisotropies and low disorder the
transition between the low-temperature solid phase and the
high-temperature liquid phase is first order. The superconducting coherence is lost simultaneously in all directions at
the melting temperature T m .
10 885
FIG. 1. Resistivities ~in arbitrary units! of a sample of 18318
318 junctions with h 2 540 and D50.7. ~a! linear plot and ~b!
logarithmic plot. Lines are a guide to the eye. Temperature is in
units of the mean Josephson energy of the in-plane junctions.
~ii! For moderate or low anisotropy and high disorder two
continuous transitions are observed for samples with finite
thickness: from a low-temperature disordered solid phase to
a disentangled liquid phase at intermediate temperatures and
finally to an entangled liquid phase at high temperatures.
~iii! For high anisotropies and moderate or high disorder
the temperature evolution of the transport properties shows a
thermally activated behavior.
In what follows we analyze the temperature dependence
of the transport properties in this last regime and calculate
activation energies for the vortex motion. The magnetic field
H used in all simulations corresponds to 1/6 flux quanta per
plaquette. This field is commensurate with the underlying
lattice of junctions, and in the absence of disorder gives a
fundamental state of the system in which vortices arrange in
an undistorted triangular lattice. However, when disorder is
included, the fundamental state is determined mainly by the
defects, and the vortex structure becomes disordered. Due to
this, commensurability effects between vortex structure and
the subjacent lattice are expected to be negligible.13
III. RESULTS
We first present results corresponding to uncorrelated disorder. With a random distribution of critical currents we represent point defects. It should be noted that the largest disorder is due to the in-plane random distribution of critical
currents: the mean interplane critical current and its dispermin
2
sion DI c 5I max
c 2Ic are a factor h smaller than the in-plane
2
ones. For large h , the pinning energy is much larger for
vortices oriented along the c axis than for vortex lines confined between two planes.
The low-temperature resistivities obtained by numerical
integration of Eqs. ~1! and ~2! are shown in Fig. 1~a! for a
sample of 18318318 junctions, h 2 540 and D50.7. With
these parameters we simulate a highly anisotropic system
with uncorrelated disorder. The temperature dependence of
the resistivities, shown in the form of an Arrenius plot in Fig.
1~b!, suggests a thermally activated behavior. For this
sample, the data indicate a slightly smaller activation energy
for the c-axis resistivity. This result is due to finite-size effects: a systematic simulation for different sample sizes
M. F. LAGUNA, E. A. JAGLA, AND C. A. BALSEIRO
10 886
FIG. 2. Size dependence of the resistivities. L c 518 is kept fixed
and L ab 512, 18, 24 for squares, circles, and triangles. Other parameters as in Fig. 1.
shows that as system size increases the activation energy for
both the in-plane and the out-of-plane resistivities approach
the same value, as can be seen in Fig. 2. Based on the scaling
of this figure we conclude that, for the parameters being
simulated and for large systems, the 3D JJA model gives the
same value for the activation energies of the two components
of the resistivity in agreement with the experimental
observation.9,10
In order to interpret the results we write the in-plane resistivity as follows:
r ab 5n
DL
,
k BT
~4!
where n is the density of mobile vortices along the c axis and
D L is their long-time diffusion constant. In our model the
number of mobile vortices, for the temperature range of interest, is a large fraction of the total number of field induced
vortices. The density n is then fixed by the external magnetic
field and the temperature dependence of the resistivity is
dominated by the temperature dependence of the diffusion
constant that reflects the thermally activated nature of the
vortex motion. The out-of-plane resistivity can be written as
r c 5n p
D pL
,
k BT
~5!
where n p is the number of vortex lines that cross the sample
perpendicularly to the c axis. As we mentioned above, these
lines are created by the entanglement of the field-induced
vortices and by the thermal excitation of small vortex loops.
The density n p is temperature dependent and the behavior of
the c-axis resistivity is determined both by the number of
percolating vortex lines and by their diffusion coefficient
D pL . It is difficult to separate the two effects. On one hand,
there is no simple algorithm to count the number of percolating lines since there is not a unique way of defining them.
On the other hand, in order to measure the diffusion coefficient one should identify one of these lines and follow it as a
function of time, a procedure that also presents numerical
problems.
To partially overcome this difficulty, we may characterize
the vortex structure by calculating, following the procedure
described in Refs. 8,13, the percolation probability P as a
function of temperature. In Fig. 3, P and the corresponding
values of the resistivity are shown for the same parameters as
in Fig. 1. The percolation probability is nearly one at all
temperatures with a shallow minimum at low temperature.
57
FIG. 3. Percolation probability and resistivities for a sample
with the same parameters as in Fig. 1.
This nonmonotonic behavior of P is due to the interplay of
thermal fluctuations and pinning. The behavior of P suggests, as we discuss in more detail below, that n p is a weak
function of the temperature and that the exponential drop of
the resistivity is due to the temperature dependence of the
diffusion coefficient D pL . The results for a sample with
lower anisotropy are shown in Fig. 4. Now P drops to zero at
a temperature T p indicating that in the low-temperature state
vortices are disentangled. When this happens the c-axis resistivity goes rapidly to zero for T,T p as shown in the figure. This vanishing of r c is associated to a rapid variation of
n p close to T p . The behavior of Fig. 3 is typical of systems
with large anisotropy. For lower anisotropy the typical behavior is the one depicted in Fig. 4.16 When the anisotropy is
progressively increased, starting from a low value, the percolation temperature moves down. At a certain critical value
this temperature changes discontinuously from a finite value
to zero, i.e., the behavior changes qualitatively from that of
Fig. 4 to the one of Fig. 3. Moreover, for anisotropies close
to this critical value we have found both behaviors by performing runs with different numerical seeds, all other parameters being identical.
Except at very high temperatures, close to the critical temperature, the percolating vortex lines threading the sample
perpendicularly to the external field cross the planes up and
down many times. Since the largest pinning energy is the
in-plane pinning, vortices that on the average cross the
sample parallel and perpendicularly to the c axis are subject
to essentially the same random potential and then have the
same diffusion coefficient. Thus, when the number of percolating vortex lines is a weak function of the temperature, the
same activation energy is obtained for the two components
of the resistivity.
In samples with columnar defects, the low-temperature
behavior of the c-axis resistivity is always dominated by the
FIG. 4. Same as Fig. 3, but with h 2 51.
57
ACTIVATION ENERGY FOR VORTEX MOTION IN . . .
FIG. 5. Resistivities for samples with c-axis-correlated disorder.
Parameters are h 2 540 and D50.7 for a sample of 18318318
junctions.
temperature dependence of n p . We have simulated columnar
defects introducing c-axis-correlated disorder in two different ways. First we present results for samples in which all
horizontal planes have the same distribution of critical currents. This type of c-axis-correlated disorder increases the
elastic constant C 66 of the vortex structure. At zero temperature, vortices are straight lines running along the field direction and forming a disordered structure in the ab plane. At
finite temperatures the thermal fluctuations generate diffusion of these lines and the in-plane resistivity shows a thermally activated behavior @Fig. 5~a!#. For the same parameter
D, the activation energy for the c-axis-correlated disorder is
larger than in the previous case since pinning is more efficient. The c-axis resistivity is vanishingly small if temperature is lower than some characteristic value T p . This is in
agreement with the fact that percolation probability P is zero
below T p , as seen in Fig. 5~b!. Only when thermal fluctuations overcome the energy barriers associated with vortex
bending, the vortex lines entangle. When this occurs the
c-axis resistivity shows a rapid increase as a function of
temperature. The characteristic temperature T p , that measures the energy scale for entanglement, is higher than the
one obtained in the case of uncorrelated disorder.
From the numerical data, obtained in relatively small
samples, it is difficult to define a large temperature interval
where the Arrenius plot clearly shows an exponential behavior for r c . However it becomes evident that, at low temperatures, the two components of the resistivity are determined
by two different energy scales. If from the numerical data we
were to define activation energies in the low-temperature regime, the activation energy D c for the c-axis resistivity
would become much larger than the corresponding D ab . In
this regime, while r ab is determined by the diffusion coefficient, the temperature behavior of r c is controlled by n p .
We have also simulated columnar defects taking a stack
of identical planes with a random distribution of nodes, with
concentration c d , for which all the junctions with the nearest
neighbors have zero critical current. As shown in Fig. 6, the
results in these cases are qualitatively similar to those obtained with c-axis-correlated random distribution of critical
currents. In the present case however, the numerical data of
Fig. 6 are consistent with an exponential decay of r c with a
10 887
FIG. 6. Resistivities for a sample with columnar defects simulated as described in the text. The concentration of defects is c p
50.2. Lines are a guide to the eye.
large activation energy D c . In systems with columnar defects, the matching field B f is defined as the field that induces a number of vortices of the order of the number of
defects. For fields B smaller than B f and low temperatures
each vortex is pinned in one defect. If B.B f a fraction of
the vortices are pinned while the rest can diffuse. Experimentally different activation energies are observed only when the
magnetic field is smaller than the matching field B f . For B
@B f the two components of the resistivity show the same
value of the activation energy.10 This fact seems to contradict
our results: the data of Fig. 6 were obtained with a number of
defects much smaller than the number of vortices, but the
temperature dependence of the resistivities cannot be characterized by the same activation energy. It should be noted
however that a columnar defect affects all vortices lying at a
distance smaller than a characteristic distance L d which is
determined by the vortex-vortex and vortex-defect interactions. To simulate the case B;B f , the average distance between defects should be much larger than L d and the number
of vortices of the order of the number of defects. In our small
samples this would imply to take a few vortices and defects
giving rise to a very poor statistics. For the parameters of
Fig. 6 all vortices are very much affected by the columnar
defects and the crossover from B,B f to B.B f is not observable.
IV. SUMMARY AND DISCUSSION
We have studied the low-temperature resistivity of highly
anisotropic superconductors in the presence of an external
magnetic field. The system consists of a stack of weakly
coupled superconducting planes with the external field perpendicular to them. In the presence of intrinsic disorder, the
resistivities for currents parallel and perpendicular to the
planes show low-temperature tails that can be interpreted as
a thermally activated behavior.
The numerical results can be summarized as follows:
~i! For point defects ~uncorrelated disorder! the activation
energies for the two components of the resistivity r ab and r c
are equal. To show this, some finite-size scaling is necessary
since finite-size effects may mask the results. In some cases
10 888
M. F. LAGUNA, E. A. JAGLA, AND C. A. BALSEIRO
~when the anisotropy is not large enough!, the vortex structure disentangles at a temperature T p . When this occurs r c
decreases faster than r ab as the temperature decreases below
T p . For highly anisotropic and thick samples and based on
previous results, we expect T p to vanish, making this last
effect of no physical relevance.
~ii! For samples with columnar defects there is no temperature range where the two components of the resistivity,
r ab and r c , can be characterized by the same energy scale.
The results of Fig. 6 clearly show a larger activation energy
for the c-axis resistivity.
We interpreted these results in terms of a simple picture in
which the resistivity is written in terms of the number of
mobile vortex lines and their diffusion constant. For r ab , the
relevant vortex lines—with density n and diffusion constant
D L —are the field-induced vortices threading the sample
along the c axis. For r c , dissipation is due to the existence
of percolating lines in the entangled vortex structure which
cross the sample perpendicularly to the c axis; their density
n p and diffusion constant D pL are temperature dependent.
We reached the conclusion that for point defects, the temperature dependence of both r ab and r c is dominated by the
diffusion constants. In our model in which the in-plane pinning is dominant, the diffusion constants D L and D pL are
characterized by the same activation energy: in the tempera-
G. Blatter et al., Rev. Mod. Phys. 66, 1125 ~1994!.
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57
ture range of interest, all vortex lines cross the planes and are
affected by the same random potential.
Columnar defects enhance the rigidity of vortices and the
percolation temperature T p at which the vortex structure entangles moves up relative to the case of point defects. This
introduces a new relevant energy scale and the two components of the resistivity cannot be characterized by a single
activation energy. While the temperature dependence of r ab
is controlled by D L , the temperature dependence of r c is
determined by n p .
Our simple picture is useful to interpret the results, but
certainly it is not complete. The diffusion constants D L and
D pL depend on the vortex structure and in general are not
independent of the variable n p . Although it is numerically
cumbersome, an independent evaluation of the diffusion coefficients and the variables n and n p would be interesting.
This would permit to confirm the validity of expressions ~4!
and ~5!, and to gain insight into the origin of the physical
processes responsible for the dissipation of the vortex structure.
ACKNOWLEDGMENTS
E.A.J. acknowledges financial support by CONICET.
C.A.B. is partially supported by CONICET.
H. Safar et al. ~unpublished!.
A. E. Koshelev, Phys. Rev. Lett. 76, 1340 ~1996!.
12
D. Domı́nguez et al., Phys. Rev. Lett. 67, 2367 ~1991!.
13
E. A. Jagla and C. A. Balseiro, Phys. Rev. B 53, 15 305
~1996!.
14
D. Domı́nguez, N. G. Jensen, and A. R. Bishop, Phys. Rev. Lett.
75, 4670 ~1995!; R. E. Hetzel, A. Sudbo, and D. A. Huse, ibid.
69, 518 ~1992!.
15
E. A. Jagla and C. A. Balseiro, Phys. Rev. Lett. 77, 1588 ~1996!.
16
For low disorder and intermediate anisotropies, a regime appears
in which the vanishing of resistivity is associated to a first-order
transition in the system ~Ref. 4!. The value D50.7 is close to
the critical disorder for this to occur.
1
10
2
11