PHYSICAL REVIEW B
VOLUME 58, NUMBER 1
1 JULY 1998-I
Oscillatory dynamics of the driven vortex solid in YBa2Cu3O72d single crystals
S. N. Gordeev,* A. P. Rassau, and P. A. J. de Groot
Department of Physics, University of Southampton, Southampton SO17 1BJ, United Kingdom
R. Gagnon and L. Taillefer
Department of Physics, McGill University, Montreal (Quebec), Canada H3A 2T8
~Received 7 November 1997!
Investigations have been carried out on the dynamics of the vortex solid in clean detwinned YBa2Cu3O72d
single crystals at temperatures and fields just below the thermodynamic melting line. It has been observed that
the dynamics of this system is crucially dependent upon the modulation form of the driving current. Very
different responses were obtained in the case of double-polarity square-wave and single-polarity square-wave
currents, and these responses were attributed to relatively ‘‘ordered’’ and ‘‘disordered’’ dynamic states, respectively. In particular, we present investigations on a type of dynamic vortex state that manifested itself as
a large amplitude oscillation in the voltage response amplitude ~V 0 varied by as much as a factor of 3!. This
oscillatory state arose when a periodically modulated drive current ~f m ;100 Hz! was applied, which was
asymmetric either in terms of the positive and negative going parts of its period or in terms of its amplitude.
We have found that the frequency of these V 0 (t) oscillations ~typically in the range 0.01–0.1 Hz! did not
depend upon the frequency of the drive but that it was proportional to the asymmetry of the current modulation.
In this paper we explore the range of fields, temperatures, applied current amplitudes, and asymmetries over
which the oscillatory response was observed. We have shown that the period of the oscillations corresponds
approximately to the time that it would take for the vortex system to drift coherently twice across the width of
the sample. Using this and other features of our observations, we propose an interpretation of the oscillatory
effects based upon the idea of defect superstructure that drifts across the sample with the same velocity as that
of individual vortices. @S0163-1829~98!03625-X#
I. INTRODUCTION
Over recent years the dynamic behavior of elastic manifolds in the presence of random frictional forces has attracted
considerable attention. This generic problem has implications for a very wide range of systems from the dynamics of
driven flux-line lattices in type-II superconductors to the motion of tectonic plates. In general, the motion of such systems
is expected to be highly disordered. However, computer
simulations based upon simple models have revealed that
under certain specific conditions organization may
develop.1,2 The observation of various memory effects in the
dynamics of driven charge-density waves has provided
strong evidence for the occurrence of such organizational
phenomena.3 Other physical systems that manifest analogous
effects include Wigner crystals of electrons,4 magnetic
bubbles systems,5 and sliding tectonic plates.6 In this paper
we present results that demonstrate that under certain conditions the driven flux-line lattice in high-T c superconductors
can also manifest a form of ‘‘self-organized’’ behavior.
There is a great deal of evidence, both theoretical and
experimental, demonstrating that the dynamics of a driven
vortex system depends crucially on the magnitude of the
driving force.7,8 It is becoming increasingly accepted that the
response of the vortex system can be classified into three
distinct dynamic regimes. At the low driving-force limit ~often known as the ‘‘elastic’’ regime!, the motion of the pinned
vortex solid is controlled by thermal activation processes and
the response of the system is Ohmic.7 For intermediate driving forces, which are comparable in magnitude to the overall
0163-1829/98/58~1!/527~11!/$15.00
PRB 58
pinning force, theory predicts that the flow of vortices will be
highly incoherent. In this so-called plastic flow regime,
which is characterized by non-Ohmic voltage responses and
high levels of voltage noise, vortex motion is expected to
occur along static channels.9–11 Even in the case of relatively
low bulk pinning, where the overall motion is predominantly
controlled by the flow of vortices across inhomogeneous surface barriers, channels are expected to develop at points
along the surface where it is easiest for the vortices to penetrate. The random opening and closing of such channels
could account for the high levels of voltage noise observed
in the plastic flow regime.12
At driving forces that are much larger than the overall
force due to pinning, a transition into a coherently moving
dynamic regime is expected.13 Evidence for such a transition
has been provided by transport experiments carried out on
NbSe2 single crystals.14 In these studies two distinct dynamic
states with very different transport properties were identified,
each of which was found to be stable in a particular region of
the H-T plane and metastable elsewhere. The dynamics of
the system at the high driving-force limit can be described in
terms of Bardeen-Stephen free flux flow of virtually all the
vortices, and pinning sites are only expected to have a very
weak perturbing effect on the vortex dynamics in this case.13
The fact that there is definitely some degree of ordering at
this limit has been demonstrated by recent neutrondiffraction experiments.15 There is, however, still considerable controversy as to the precise nature of this ordering.10,16
Our recent transport experiments17 have shown that, in
addition to the dependence of the dynamics on the magnitude
527
© 1998 The American Physical Society
528
S. N. GORDEEV et al.
of the applied current, the form of the current modulation is
also important. We have observed a large increase in the
voltage response amplitude accompanied by a substantial decrease in the voltage noise when a double-polarity squarewave ~DPSW! current is applied instead of a direct current
~dc! or single polarity square wave ~SPSW!. This suggests
that when vortices move backwards and forwards, with no
net motion, this leads to ordering of the vortex lattice. In
order to study the effect of varying the net velocity of the
system, we proposed a method utilizing asymmetrically
modulated square-wave currents. By changing the modulation asymmetry we were able to gradually vary the net vortex velocity while keeping the amplitude of the driving force
constant. Using this asymmetric modulation method we discovered a type of the vortex dynamics.18,19 For relatively
small applied current asymmetries (;0.05– 0.1 %) we observed a slow periodic variation with time of the voltage
response amplitude V 0 . Kwok et al.20 have recently observed similar oscillatory effects.
In this paper we present the results of our studies on the
influence of the applied current magnitude, asymmetry, and
modulation type on the V 0 (t) oscillations. We also consider
temperature and field-related effects. Importantly, we show
that the period of the oscillations corresponds approximately
to the time taken for the vortices to drift twice across the
width of the sample. This observation, together with other
features of our data, leads us to propose a phenomenological
interpretation of the oscillations, based upon the idea of a
defect superstructure that drifts with the same net velocity as
the vortices across the width of the sample. We suggest that
this defect superstructure consists of two distinct regions
characterized by different degrees of relative disorder. In the
framework of this interpretation, periodic changes in the proportion of the two dynamic phases give rise to corresponding
changes in the overall mobility of the vortex system, thereby
leading to oscillations of the output voltage response.
II. EXPERIMENTAL DETAILS
The effects described in this paper have been observed on
a number of different YBa2Cu3O72d single crystals, all
of which were grown using a self-flux method in an Ystabilized ZrO2 crucible.21 Oscillatory voltage responses
were, however, only obtained in the case of detwinned
samples with sufficiently low defect densities. For consistency, all of data presented in this paper were obtained on
one particular sample of dimensions 1.12 mm, 0.35 mm, and
65 mm along a, b, and c crystalline axes, respectively. This
sample was oxygenated for 10 days at 500 °C in flowing O2
and quenched at room temperature. It was then detwinned by
applying a uniaxial pressure of 50 MPa at 550 °C for 30 min.
Finally it was reoxygenated for 1 day at 500 °C in an O2
atmosphere. No twin planes were visible on examining the
crystal using polarized light microscopy, although this does
not guarantee the absence of twins on the submicrometer
level. However, since twin-boundary pinning leads to a pronounced dip in R( u ) dependence22 at u 50° ~where u is the
angle between the applied field and the crystalline c axis!,
angular-resolved magnetoresistance measurements allow us
to test for the presence of twin planes. For the sample used in
this particular study there was no such dip in the R( u ) de-
PRB 58
pendence at u 50°, thereby providing evidence for the absence of microtwins.
Current and voltage contacts were painted onto the
sample using silver epoxy ‘‘Du Pont 6838’’ and subsequently annealed at 500 °C in an oxygen atmosphere. The
current contacts were formed on the sides of the crystal so as
to ensure a uniform current along the a axis. The pads for the
voltage leads ~two parallel strips! were painted onto the top
surface. The resistance of the contacts made in this way was
approximately 1 V. Magnetoresistance measurements were
performed using a superconducting quantum interference device picovoltmeter with a comparatively broad frequency
bandwidth ~from 0 up to 3 kHz! that allowed us to control
the shape of the voltage response by monitoring the output
wave form as displayed on the screen of an oscilloscope.
Voltage signals were measured using a conventional lock-in
amplification technique. Magnetic fields up to 5 T ~with
H i c! were generated using a superconducting solenoid operating in the persistent current mode. The temperature stability during isothermal measurements was better than 2 mK.
The sample was found to have a zero-field resistivity transition width of 0.3 K at the critical temperature T c 593.6 K
and the resistivity along the a axis at T5100 K was measured as r a 564 m V cm. Unless otherwise indicated, measurements were performed at B52 T and T587 K, which
lies 2 K below the thermodynamic melting temperature for
this field. The results presented in this paper therefore correspond to the dynamic behavior of the vortex solid.
To determine the influence of thermal effects on our data
we measured r (T) curves in the region of the melting transition for a range of different currents. It was found that the
sharp onset of the resistivity kink ~conventionally associated
with the vortex solid-to-liquid melting transition! occurred at
the same temperature for all currents of up to 20 mA. Thus,
since the largest current that we used in studies was 5 mA
~corresponding to a current density of 22 A/cm2!, this implies that our observations cannot be due to thermal effects.
III. RESULTS
A. Oscillations of the voltage response amplitude
As was shown in our previous papers,17–19 under certain
conditions an asymmetrically modulated probing current can
induce periodic variations in the amplitude of the voltage
response V 0 . A typical example of such V 0 (t) oscillations is
shown in Fig. 1~a!. In this case, the applied current had an
amplitude I 0 53 mA and was modulated by an asymmetric
square wave ~ASW! with frequency f m 568 Hz and ‘‘temporal’’ asymmetry a t 5(T 1 2T 2 )/(T 1 1T 2 )50.1. Here T 1
and T 2 represent the durations of the positive and negative
going parts of the square-wave period, respectively. In general, the time dependence of the voltage response V(t) approximately reproduced the shape of the applied ASW current @see the schematic representation in Fig. 1~b!#. However,
while the amplitude of the applied current was constant, the
voltage response amplitude (V 0 ) was seen to execute slow
periodic variations with time. Notice, that the frequency f osc
of these V 0 (t) oscillations (;0.01 Hz) was much lower than
the modulation frequency f m 568 Hz of the driving current.
Furthermore, as we will show below, f osc was independent of
the modulation frequency for f m .40 Hz. In general, the
PRB 58
OSCILLATORY DYNAMICS OF THE DRIVEN VORTEX . . .
FIG. 1. ~a! Typical example of the voltage amplitude (V 0 ) oscillations obtained using a driving current modulated with an asymmetric square wave ~ASW!. The driving current was modulated
with a frequency f m 568 Hz, had amplitude I 0 53 mA and temporal
asymmetry a t 50.1 ~as defined in the text!. ~b! Schematic representation of the voltage response (V) obtained using an ASW current
as it would be observed on the screen of a monitoring oscilloscope.
The amplitude of this response (V 0 ) varies with time @it is the
envelope of this variation that is depicted in ~a!#. Note that for
clarity of presentation the square-wave oscillations shown have a
frequency ;1 Hz. In reality the typical modulation frequencies
were of the order of 100 Hz; thus one oscillation period would
consist of about 4000 voltage pulses.
shape of the observed V 0 (t) oscillations was asymmetric
such that the rising edge of each oscillation cycle had a shallower gradient than the falling edge. While the falling edges
were approximately linear, the rising edges tended to be less
smooth, showing a fine structure that often repeated itself
across a number of oscillation cycles.
Shown in Fig. 2 are the time dependences of the voltage
response for three different applied-current modulation
forms with the same applied-current amplitude (I 0
53 mA). All other parameters are the same as for the curves
presented in Fig. 1. It can be seen from Fig. 2 that very
different voltage responses were obtained for these different
modulation forms. In the case of the SPSW modulation the
time-averaged voltage response amplitude was low and associated with high levels of voltage noise. For the DPSW
current, V 0 was relatively high and noise levels were much
lower than in the SPSW case. The amplitude of the response
to the ASW driving current varied periodically between upper and lower limits ~V max
and V min
0
0 ! that lay between the
responses obtained in the case of SPSW and DPSW drive
min
currents. We have found that the ratio V max
0 /V0 is dependent
upon the probing current amplitude ~the maximum value
min
which we observed was V max
0 /V0 '3!. For the case shown in
max min
Fig. 2 the ratio V 0 /V0 '2.
B. B-T window for oscillatory effects
V 0 (t) oscillations similar to those shown in Fig. 1~a! were
observed over a limited temperature range the width of
which was dependent on both the magnitude of the applied
529
FIG. 2. Time dependence of the voltage response amplitude
(V 0 ) for different applied current modulation forms, namely,
double-polarity square wave ~DPSW!, single-polarity square wave
~SPSW!, and asymmetric square wave ~ASW!. In all cases the applied current had an amplitude I 0 53 mA and was modulated with a
frequency f m 568 Hz. In the case of the ASW modulation the temporal asymmetry parameter a t 50.1. Notice the transient response
on switching from DPSW to ASW modulations. It can be seen that
in the ASW case V 0 varies between limits that lie between the
time-averaged responses for the DPSW and SPSW drives ~as represented by the dashed lines!.
magnetic field and the amplitude of the ASW probing current. To demonstrate the influence of the oscillations on the
R(T) curves we have compared the temperature dependence
of the resistance for different types of probing current modulation ~see Fig. 3!. In contrast to the ASW response, no os-
FIG. 3. Resistance vs temperature curves obtained with different
applied-current modulation forms and amplitudes. Curves A – C
were obtained using SPSW current modulations with appliedcurrent amplitudes I 0 50.1, 1.0, and 3.0 mA, respectively ~in each
case they were obtained by cooling at a rate of 0.2 K/min!. Curves
D and E were both obtained using an ASW current with amplitude
I 0 53 mA and asymmetry a t 50.03 but with different cooling rates
of 0.2 and 0.1 K/min, respectively. Notice in particular the difference between curve C and curves D and E, which were obtained
with the same amplitude of applied current but with different modulation forms.
530
S. N. GORDEEV et al.
cillations were observed in the case of SPSW modulation.
Correspondingly, the R(T) curves had a form that is typical
of clean detwinned YBa2Cu3O72d crystals.23 Shown in Fig. 3
are the R(T) dependences obtained using a SPSW current
modulation (B52 T) for three different applied current amplitudes I 0 50.1, 1.0, and 3.0 mA ~curves A, B, and C, respectively!. The sharp drop in the resistance at T m 589 K is
commonly taken as evidence of a first-order phase transition
from a vortex liquid state to a vortex solid state.23 Below T m
the Ohmic resistance dropped abruptly ~over a temperature
interval <0.1 K! by several orders of magnitude, becoming
immeasurably small at lower temperatures. For higher I 0 values ~curves B and C! the transition broadened strongly and a
pronounced minimum appeared in the R(T) dependence at a
temperature T p that lies 0.25 K below T m . This minimum,
which is often observed near to the melting line, has been
associated with the peak effect ~PE!.24 The anomalous behavior associated with the PE has been explained in terms of
a premelting softening of the vortex lattice that leads to an
increase in the effective pinning force.25 Below T p the voltage response showed large irreproducible fluctuations that
were much larger than the base noise levels associated with
the picovoltmeter arrangement. This seems to indicate that in
this regime the flow of vortices is highly nonuniform.
Above T m the form of the R(T) curves was found to be
independent of the particular modulation form. In contrast,
below T m a pronounced difference developed between the
curves obtained using different modulation forms. Curves D
and E of Fig. 3 were obtained using ASW probing currents
with the same amplitude (I 0 53 mA) and asymmetry ( a t
50.03) but with different cooling rates of 0.2 and 0.1 K/min,
respectively. Above a resistance level of approximately
331025 V both of these curves are highly nonmonotonic,
consisting of a series of unequally spaced maxima and
minima. This nonmonotonicity is a manifestation of the same
effect that gave rise to the oscillations in the V 0 (t) dependences ~as shown in Figs. 1 and 2!. Curve E ~obtained with
a cooling rate dT/dt50.1 K/min! shows approximately
twice as many peaks as curve D ~obtained with a cooling rate
dT/dt50.2 K/min!. On comparing the various results presented in Fig. 3 it is clear that the form of the curves in the
ASW case is due to the superposition of a monotonic
temperature-dependent component and a time-dependent oscillatory component. Curves D and E show the same monotonic trend, reflecting the temperature dependence of the resistance, but the oscillatory components differ due to the
different sweep rates.
An important distinction between the curves obtained using the different modulation forms is that, for the same applied current amplitude, the resistance ~at a given temperature T,T p ! was always found to be higher in the ASW case.
In addition, the R(T) dependences obtained using an ASW
drive were always much less noisy. These observations indicate that ASW modulation leads to a more ordered flux-flow
regime together with a higher overall vortex mobility. Figure
4 shows R(T) dependences similar to curves D and E of Fig.
3, but obtained with three different amplitudes of applied
ASW current ~at a cooling rate of 0.2 K/min!. From Fig. 4 it
is clear that oscillations only appeared for higher I 0 values
where the effect of the pinning wells was sufficiently suppressed by the applied current. For small ASW current am-
PRB 58
FIG. 4. Resistance vs temperature curves obtained on cooling at
a rate 0.2 K/min for three different amplitudes of applied ASW
current. In each case the ASW modulation had asymmetry a t
50.03.
plitudes (I 0 ,1.5 mA) the resistance was seen to drop monotonically to zero, the response in this case being
indistinguishable from that obtained using SPSW or DPSW
current modulations.
Shown in Fig. 5 are R(T) oscillations, obtained on driving
the system with an ASW current ~I 0 53 mA and a t 50.03!
for different magnitudes of applied magnetic field. As B was
increased, the R(T) curves shifted to lower temperatures following the decrease in the melting temperature T m . It can be
seen from this figure that the temperature interval between
successive minima decreased with increasing R and in gen-
FIG. 5. Resistance vs temperature curves ~cooling rate 0.2
K/min! obtained for different values of applied magnetic field. Notice that for any given field, the temperature interval between successive minima along the curves decreases with increasing R. Since
all of the curves were obtained at the same constant cooling rate,
this spacing can be used as a measure of the period of the underlying V 0 (t) oscillations. Inset: Variation of the period of the underlying oscillations with applied magnetic field. The points represented by open circles were deduced from the spacing of the first
two minima along the R(T) curves in the main frame of the figure
whereas those represented by closed circles were extracted from the
spacing of the second and third minima.
PRB 58
OSCILLATORY DYNAMICS OF THE DRIVEN VORTEX . . .
FIG. 6. Voltage amplitude oscillations obtained using various
types of ~asymmetric! applied-current modulation ~for clarity the
curves have been vertically shifted!. Curves A and B were obtained
using square waves which were asymmetric in terms of the their
period ( a t 50.1) and their amplitude ( a a 5I dc /I 0 50.025), respectively. Curves C and D were obtained in response to, respectively,
asymmetrically modulated sinusoidal and triangular driving currents. In the case of curves C and D, the asymmetry was achieved
by superimposing a dc offset (I dc50.25 mA) onto a pure oscillatory
wave form of amplitude I 0 55 mA. Note that the curves in this
figure have been vertically shifted for clarity.
eral that the maxima and minima were more closely spaced
for lower magnetic fields. Changes in the spacing of the
minima reflect underlying changes in the period t osc of the
associated V 0 (t) oscillations. Thus, since all of the curves
were measured at the same ~constant! cooling rate, we can
use this spacing to estimate the period of the underlying
V 0 (t) oscillations. The inset of Fig. 5 shows how t osc , as
estimated from the spacing of minima along R(T) curves,
varied with applied magnetic field. The open circles were
extracted from the spacing of the first and second minima at
the low-temperature end of the R(T) curves whereas the
closed circles were extracted from the second and third
minima. In both cases the extracted t osc(B) dependences can
be seen to be linear.
C. Effect of applied-current parameters upon the oscillations
1. Current modulation form
As shown in Fig. 6, we have found that V 0 (t) oscillations
similar to those shown in Fig. 1 can be obtained for different
types of the applied-current asymmetry and modulation.
Curve A of Fig. 6 is similar to that shown in Fig. 1~a! and
corresponds to a square-wave current with a ‘‘temporal’’
asymmetry a t 50.1. Curve B was obtained in response to a
square-wave current that had an asymmetry in its amplitude.
This type of asymmetry was produced by superimposing a
small dc offset (I dc) onto a DPSW current with amplitude
I 0 ; thus we define the amplitude asymmetry parameter as
a a 5I dc /I 0 ~a a 50.025 for the data represented by curve B!.
It can be seen from Fig. 6 that the oscillations that arise due
to a drive with an amplitude asymmetry have practically the
same shape as those obtained using a ‘‘temporal’’ asymme-
531
try. We will show below that the oscillations are more sensitive to the amplitude asymmetries than to temporal asymmetries.
Modulation by a square wave has one important disadvantage: The reaction of the vortex system and measurement
electronics to the sharp fronts of the probing current can be
quite complicated. To show that the fronts do not play a role
in establishing the V 0 (t) oscillations we have investigated
the effect of applying asymmetric sinusoidal ~curve C, Fig.
6! and triangular ~curve D, Fig. 6! current modulations. In
both cases the asymmetry was produced by superimposing a
direct current (I dc50.25 mA) onto a pure sinusoidal or triangular wave form ~amplitude I 0 55 mA! thus giving an
asymmetry parameter a a 50.05 ( a a 5I dc /I 0 ). Voltage amplitude oscillations were observed in both of these cases, but
they were not as regular as those obtained in response to an
ASW drive. This difference can be attributed to the fact that
for triangular and sinusoidal modulations the interaction of
the vortices with the pinning centers ~and therefore their average velocity! varies significantly over any given modulation period. From this point of view, it is much easier to
analyze the behavior of vortices in the case of a square-wave
current with a ‘‘temporal’’ asymmetry: For this form of
modulation the Lorentz force has a constant value and the
macroscopically averaged interaction of the vortices with the
bulk pinning centers is not expected to vary during any given
current pulse.
The results presented in Fig. 6 clearly demonstrate that
the precise shape of the alternating current is not crucial in
establishing V 0 (t) oscillations. It should be noted, however,
that we did not observe oscillations when an SPSW current
was applied. This seems to indicate that the backwardsforwards motion associated with the alternating drive is crucial in producing these oscillatory effects. While the general
shape of the oscillations was the same for the different types
of alternating modulation, the most perfect oscillations were
obtained using ASW currents. For this reason, all of the results presented in the following sections were obtained using
ASW currents with temporal asymmetries.
2. Applied-current amplitude
Figure 7 shows how V 0 (t) oscillations, obtained using an
ASW current ~with temporal asymmetry a t 50.1!, evolved as
the applied-current amplitude was varied. The measurements
were carried out at constant field B52 T and temperature
T587 K. The amplitude of the voltage response V 0 was
measured while increasing the ASW current amplitude at a
constant rate of 1026 A/s. The curve obtained in this way
represents the superposition of a slow monotonic increase in
the mean voltage ^ V & onto a V 0 (t) oscillation. Significantly,
the oscillations were observed to appear abruptly at a particular threshold current I t 52.45 mA. Above this point, the
period of the oscillations decreased as the current was
ramped. Furthermore, it can be seen that the oscillations became increasingly erratic at higher currents; thus there exists
a relatively narrow current window over which the conditions for establishing the oscillations are optimal.
3. Magnitude of modulation asymmetry
Figure 8 presents V 0 (t) oscillations obtained using an
ASW current ~amplitude I 0 53 mA! with different values of
532
S. N. GORDEEV et al.
FIG. 7. Voltage response variation obtained on increasing the
amplitude of an asymmetrically modulated ( a t 50.1) drive current
at a rate of 1026 A/s. Notice in particular that oscillations appear
abruptly at a threshold current I t 52.45 mA.
‘‘temporal’’ asymmetry a t . Where the modulation was symmetrical ( a t 50), the amplitude of the voltage response was
constant and practically the same response was observed for
very small asymmetries. Voltage amplitude oscillations only
appeared for asymmetries in excess of a critical value a cr
'0.01 and the oscillations were most regular for asymmetries in the range 0.03, a t ,0.16. Shown in Fig. 9 are Fourier power spectra derived from the curves presented in Fig.
8 ~the spectrum for a 50 has been omitted since there was
clearly no oscillatory response in this case!. These spectra
show clear peaks at a fundamental frequency and several
smaller peaks at the harmonics. The appearance of harmonics is a reflection of the nonsinusoidal nature of the V 0 (t)
dependences. The data points represented by the open circles
in Fig. 10 show how the fundamental frequency f ocs of the
oscillations ~extracted from the Fourier spectra! varied as a
function of a t . It can be seen that f ocs increased linearly with
FIG. 8. Voltage response oscillations obtained using asymmetrically modulated square-wave currents with a range different temporal asymmetries a t ~for clarity the curves have been vertically
shifted!. In all cases amplitude I 0 53 mA. Note that a t 50 corresponds to a symmetric double-polarity square-wave current.
PRB 58
FIG. 9. Fast-Fourier transform power spectra obtained from the
curves shown in Fig. 8 ~the spectrum obtained from DPSW response has been omitted since there were clearly no periodic components in this case!. Notice how the shape and position of the
fundamental peaks in these spectra change as the applied-current
asymmetry is increased ~for clarity of presentation the spectra have
been vertically shifted and the curves for a 50.2 and a 50.4 have
been scaled by factors of 2 and 20, respectively!.
increasing a t over the approximate range 0.04–0.16. For
a t .0.16 the f ocs( a t ) dependence starts to deviate from a
linear form. This corresponds to the point at which the V 0 (t)
curves become increasingly erratic and is seen as a decrease
in the height of the peaks in the Fourier power spectra ~Fig.
9!. The more erratic the oscillations became, the larger was
the deviation of f osc( a t ) from a linear dependence. For
asymmetries above about 0.4, the V 0 (t) dependences became
very irregular and the peaks in the Fourier spectra disappeared altogether.
We have also studied how the oscillations evolved as the
‘‘amplitude asymmetry’’ parameter a a was varied. As in the
case of temporal asymmetries, the fast-Fourier transforms of
the V 0 (t) curves revealed that the fundamental frequency of
the oscillations ( f osc) increased linearly with the magnitude
of the asymmetry ~closed circles in Fig. 10!. However, the
gradient of the two dependences differed by a factor of 4.5.
This difference is due to the fact that f osc depends only indirectly upon the asymmetry of the driving current. In the case
of a temporal asymmetry, the asymmetry of the voltage response is identical to that of the driving current. However,
for amplitude asymmetries, due to the nonlinear V(I) dependence, the amplitude asymmetry parameter of the voltage
response a Va 5V dc /V ac is larger than that of driving current
a Ia 5I dc /I ac ~elsewhere we have written a Ia more simply as
a a !. In the case of a power-law voltage-current characteristic
(V}I n ) the relation between the asymmetries can be written
as a Va 'n a Ia . For the data presented in Fig. 10, the V(I)
dependence for currents in the region of 3 mA can be approximated to a power-law dependence with n'4. This
value of n is in reasonable agreement with the ratio of the
gradients of the f osc( a a ) and f osc( a t ) dependences ~see Fig.
10!. It should be noted, however, that the oscillatory component of the dependences makes it difficult to extract a precise value for the power index (n) from the V(I) curves.
PRB 58
OSCILLATORY DYNAMICS OF THE DRIVEN VORTEX . . .
FIG. 10. Variation of the period of the voltage response oscillations ~extracted from the fundamental peaks of spectra like those
shown in Fig. 9! as a function of applied modulation asymmetry.
The open and closed circles represent oscillations obtained using
temporal and amplitude asymmetries, respectively.
Notwithstanding this limitation, we conclude that the fundamental frequency of V 0 (t) oscillations is directly related to
the net vortex velocity, of which the voltage asymmetry parameter is a measure.
The effective amplification of the applied asymmetry due
to the strongly nonlinear V(I) dependences can lead to a
situation whereby, over a certain range of fields and temperatures, a nominally symmetric current with a tiny parasitic
amplitude asymmetry can lead to a relatively large response
voltage asymmetry. Thus oscillations might be observed,
even if the applied current was thought to be symmetric. This
effect is particularly pronounced at low temperatures where
the V(I) dependence power-law index of YBa2Cu3O72d
single crystals can be as high as 20. For a sample at a temperature and field such that n is equal to 10, an oscillatory
voltage response would be observed if the current source has
parasitic offset of just 0.1%: This corresponds to a response
voltage asymmetry of 0.01, the lower limit at which oscillations were observed.
4. Modulation frequency
We have seen that the amplitude of the voltage response
as well as the voltage noise are quite different in the cases of
dc and ASW currents. It naturally follows that the difference
should disappear at the limit of low modulation frequency.
Figure 11 shows V 0 (t) dependences for three different frequencies of ASW current. In general, it was found that for
modulation frequencies in the range 40 Hz, f m ,200 Hz,
the V 0 (t) dependence manifested oscillations whose period
and shape were independent of f m . At lower modulation
frequencies, the V 0 (t) dependences became increasingly erratic and for f m ,4 Hz the oscillations disappeared altogether, such that the voltage response was virtually indistinguishable from that obtained in the dc case.
533
FIG. 11. Above a certain limit the period of the voltage response
oscillations is unaffected by the frequency ( f m ) of the driving current modulation. The oscillations shown were obtained using an
ASW current with a t 50.1 and I 0 53 mA.
highly unusual dynamic behavior. The voltage response was
found to be dependent on the type of applied-current modulation: While dc or SPSW currents induced nonuniform flux
flow, characterized by high levels of voltage noise, the shaking of the vortex system that resulted from a DPSW current
led to suppression of voltage noise together with an increase
in the overall voltage response. The reduced noise levels
suggest that the motion of vortices becomes more uniform
where they move backwards and forwards with zero net velocity ( v net).
In the case of ASW currents, even more unusual effects
were observed. Over a short time scale ~t,1/f m for f m
.40 Hz!, the voltage response reproduced the time dependence of the applied current. However, over longer time
scales (101 – 103 s) the amplitude of the voltage response
showed a slow period variation with time. The frequency of
these voltage response oscillations was found to be proportional to the time-averaged voltage amplitude ^ V 0 & , 18 the
asymmetry a of the response voltage, and the reciprocal of
the applied field 1/B. Therefore, since v net} a t ^ V 0 & /B, we
deduce that f osc} v net . Experiments with variable appliedcurrent asymmetries have shown that the V 0 (t) oscillations
only appear for a limited range of a t values and equivalently
over a certain window of v net values. When v net was low
( a t ,0.01) the vortex dynamics was the same as in the case
of symmetric DPSW currents. For large v net ( a t ,0.4) the
system manifested the same behavior as in the case of dc or
SPSW currents. Oscillations were only observed for ASW
currents with intermediate a t values. We have also seen that
the oscillations only appeared when sufficiently large probing currents were applied, which implies that the driving
Lorentz force must exceed some threshold value in order for
the effect to occur. One final point to note is that as shown
by the data presented in Fig. 5, there was a minimum resistance level (R min;531025 ) below which oscillations were
not observed. This implies there exists a minimum velocity
v cr at which the vortices must be driven in order for voltage
oscillation effects to occur.
D. Summary of experimental results
IV. DISCUSSION
The vortex solid in untwinned YBa2Cu3O72d single crystals with weak bulk pinning has been observed to display
The voltage response of a superconductor in the mixed
state can be related to the total number of vortices in the
534
PRB 58
S. N. GORDEEV et al.
sample N and their average velocity ^ v & 5 ( i v i /N via the
expression
V5N ^ v & F 0 /w,
~1!
where w is the width of the sample and F 0 is the flux quantum. It follows from Eq. ~1! that, independent of the detailed
mechanism involved, the oscillations that we have observed
must be caused by variations in time of either the quantity of
vortices in the sample (N) or of their average velocity ( ^ v & ).
In our experiments we have found that the ratio of the maximin
mum to the minimum voltage responses V max
0 /V0 can be as
large as 3; thus in order to account for the oscillations either
N or ^ v & must vary by the same amount. In fact, we will now
demonstrate that the maximum possible variation in N that
could occur over the course of our experiments was some
orders of magnitude smaller than this.
The internal magnetic field in the sample can differ from
the external field by as much as the self-field B sf generated
by the combination of the shielding current I sh and transport
current I. This can be estimated using the formula B sf
' m 0 (I1I sh)/w. 26 The highest probing current used in our
experiments was I55 mA and the shielding current ~as estimated from dc magnetization data! did not exceed 5 mA.
Using these upper current values we obtain a value for the
self-field, B sf;1024 T. Since our measurements were performed in fields in excess of 1 T, this implies that N was
effectively constant. It follows that the observed oscillations
of the voltage response must be due to periodic variations in
the average vortex velocity.
There are two obvious ways in which variations in the
vortex velocity could lead to commensurate variations in the
output voltage response. One possibility is that the oscillations could reflect changes in the number of vortices that
were actually moving, which would imply a highly nonuniform flux-flow regime. Alternatively, the oscillation effects
could be due to changes in the velocity of the vortex system
as a whole, which would imply that the overall motion of the
vortex system was coherent. In the latter case, the transit
time for vortices, moving across a sample of width w, would
be given by t tr5w/ v net , where v net is the net drift velocity of
the vortices. For an ASW current with temporal asymmetry
a t this transit time can be expressed as
t tr5Blw/ ~ a t ^ V 0 & ! ,
~2!
where l is the distance between voltage contacts and ^ V 0 & is
the mean voltage response ~evaluated by integration over one
period of the voltage oscillations!. Note that in arriving at
this formula, we have assumed the voltage response has the
same asymmetry a t as the applied-current modulation.
In Table I we present t tr values calculated from experimental data @using Eq. ~2!# for various different parameter
(I, a t , f m , B, T! regimes in which clear oscillations ~with
period t osc and mean value ^ V 0 & ! were observed. From the
table it can be seen that both t osc and t tr varied substantially
between the various regimes. However, in all cases the ratio
t osc / t tr was approximately equal to 2. We believe that this
constant value for the ratio cannot be fortuitous, especially
given the range and diversity of parameters considered. It is
clear, therefore, that any attempt to interpret the oscillatory
effects must also be able to explain the value of this ratio. In
TABLE I. Values of the measured oscillation period ( t osc), calculated transit time ( t tr), and the ratio of the two for a range of
different parameter values ( t osc / t tr).
I
~mA!
at
fm
~Hz!
B
~T!
T
~K!
^ V 0&
~mV!
t osc
~s!
t tr
~s!
t osc / t tr
3
3
3
4
5
3
0.10
0.10
0.04
0.10
0.10
0.10
68
128
68
68
68
68
2
2
2
2
2
3
87.0
87.0
87.0
86.4
87.0
86.0
0.108
0.103
0.126
0.147
0.262
0.089
29.3
29.1
84.1
24.5
9.90
59.2
14.0
13.9
40.2
11.0
4.48
27.2
2.10
2.09
2.09
2.23
2.21
2.18
particular, as will be outlined below, this observation allows
us to reject any interpretation based upon a model of nonuniform flux flow.
One example of how nonuniform flux flow could arise is
provided by the channeling interpretation as proposed by
D’Anna et al.12 in order to account for their observations of
the vortex system in the plastic flow regime. On measuring
the voltage response of an untwinned YBa2Cu3O72d crystal
in the geometry J i ab, H i ab (J'H) they observed sharp
peaks in the voltage noise spectrum that they associated with
the motion of vortices along channels between static solid
domains. However, from Eq. ~1! it is clear that if the quantity
of vortices that were actually moving (N m ) was a small fraction of the total number of vortices in the sample (N), then
in order to produce the same voltage response the vortices
traversing the channels would have to move with higher velocities than in the case of coherent motion. Thus the time
taken for any given vortex to traverse the length of a channel
( t ch) would be much shorter than the oscillation periods
( t osc) observed in our experiments. It should be apparent that
in order to account for the oscillatory effects, it is necessary
to identify a time scale comparable to the period of the oscillations. To arrive at an ‘‘effective’’ transit time of the
same order as t osc in the framework of the channeling interpretation it would be necessary to introduce an exotic mechanism involving the slow coherent opening and closing of
channels. It is, however, difficult to imagine how such a
model could account for the remarkable correspondence between t osc and t tr . In the discussion that follows we will
assume that the vortex system moves as a coherent whole,
such that its motion can be characterized by a single overall
drift velocity. Our interpretation of the oscillatory effects is
based upon the idea that the modulation of the voltage response reflects periodic variations in the relative disorder of
this system and thus in its overall mobility. Using this interpretation we have been able to provide an explanation for the
correspondence between t tr and t osc .
We have seen that in response to a pure oscillatory
~DPSW! drive, the vortex system exhibits much lower noise
levels than in the case of a unidirectional ~SPSW! drive ~see
Fig. 2!. This strongly suggests that these two cases represent
distinct dynamic states, characterized by different degrees of
relative disorder. Henceforth we shall refer to these regimes
as the ‘‘ordered’’ and ‘‘disordered’’ states, respectively.
However, it is not immediately obvious how this distinction
should arise: Even in a defect-free system, the flow of vortices across the surface of a sample is expected to be
PRB 58
OSCILLATORY DYNAMICS OF THE DRIVEN VORTEX . . .
disordered27 and surface imperfections ~present on the surface of all real samples! can only be expected to increase the
nonuniformity of this flow. We suggest that the essential
distinction between the DPSW and SPSW driven states is the
way in which the vortex system ‘‘heals’’ after its disorderly
flow across the surface.28,29 In the case of a unidirectional
drive, the system is driven with a relatively high velocity and
in a single direction. As such, the ‘‘disordered state’’ generated as vortices are ‘‘torn’’ across the surface will be given
little or no opportunity to heal. For an oscillatory drive, however, the repeated backwards and forwards motion of the
vortices about their equilibrium positions could allow a
gradual ‘‘healing’’ of the vortex lattice into a relatively ‘‘ordered’’ state. Thus, overall, the motion of the vortices in the
DPSW case would be expected to be much more coherent.
Another important distinction between these two dynamic
states is their very different mobilities, as reflected by the
disparity between the mean voltage responses ~see Fig. 2!.
The relatively low time-averaged voltage response obtained
in the SPSW case indicates that the ‘‘disordered’’ dynamic
state has much lower overall mobility. This can be readily
understood when we consider that in this case it is better able
to accommodate variations in the bulk pinning potential and
thus interacts more strongly with the pinning centers.
An ASW drive can be regarded as lying somewhere between the extremes of DPSW and SPSW modulations. This
seems to hint that the ASW driven state is in some way
intermediate between the ‘‘ordered’’ and ‘‘disordered’’
states that arise in the case of pure oscillatory and direct
drives, respectively. Some support for this idea is provided
by the fact that, as shown in Fig. 2, the voltage response for
the ASW driven states oscillates between maximum and
minimum levels that lie between the responses for the DPSW
and SPSW signals for the same applied-current amplitude. In
order to account for the oscillations in terms of a ‘‘disorder
model,’’ it is necessary to explain how the state of the system could change periodically in such a way that it was
always constrained between these two extremes.
There are two obvious ways in which the overall disorder
of the system could change with time. One possibility is that
the relative disorder of the system as a whole could evolve
with time. It is, however, difficult to see how this could lead
to a feedback mechanism, a necessary feature of any explanation of the oscillations. Alternatively, a situation could
arise whereby ordered and disordered states ~that could be
likened to those obtained in response to DPSW and SPSW
drives, respectively! coexisted within the sample in regions
that were separated by well-defined boundaries. In this case
the variation of the overall disorder of the system would be
due to changes in the proportion of these two dynamic states,
with the maxima and minima in the oscillations representing
the points along the oscillation cycle at which the system
was wholly in one or other of these two states. Both
experimental15,30 and theoretical31,32 studies have provided
evidence that the transition between ordered and disordered
dynamic states is hysteretic. Where hysteresis arises this is
expected to lead to an inherent stability of states that allows
for their coexistence over a particular region of parameter
space; thus a dual-state dynamic regime is at least theoretically possible. In the interpretation that follows we will assume that under the influence of an ASW drive, the system
535
consists of two distinct dynamic states separated by a welldefined boundary. Since nucleation occurs along the surface
of the sample that lies parallel to the applied current (I), it
seems reasonable to suggest that this state boundary would
also be parallel to I.
If the vortex system was divided into two well-defined
regions, as outlined above, then the periodicity of the voltage
response could be due to the motion of the boundary between these regions. We suggest that under the influence of
an ASW drive this boundary drifts across the sample with a
net velocity equal to the drift velocity of the individual vortices. One implication of this interpretation is that in the
disordered regions dislocations of the vortex lattice must
move across the sample with the same velocity as that of the
vortex system itself. Numerical simulations28 have indicated
that, at least under certain conditions, the dislocation superstructure may remain static. However, a recent numerical
simulation by Scheidl and Vinokur33 showed that the relative
velocity of the vortices and the defect superstructure may
decrease with increasing driving force and that there may
even be transition at a critical driving velocity to a comoving
regime in which dislocations move together with vortices.
Since the instantaneous velocity is expected to be high for
the relatively large current amplitudes used in our experiments, it seems reasonable that our measurements should
correspond to the high-velocity comoving regime as identified by Scheidl and Vinokur. Moreover, the fact that we have
observed a threshold current magnitude below which the oscillations were not observed could reflect the crossover to a
comoving regime.
In the framework of the two-state interpretation the transit
time @calculated using Eq. ~2!# can be associated with the
time taken for the state boundary to move across the width of
the sample. If we arbitrarily assume that a given oscillation
cycle starts with the system in a completely ordered state,
then we can conveniently divide the oscillation cycle into
two parts. Over the first part, the cycle of the proportion of
the system in the disordered state would gradually increase
until it was wholly disordered. As the proportion of the two
dynamics states changed the boundary separating them
would move across the width of the sample. It should, therefore, be clear that over the course of any given oscillation
cycle the boundary between dynamic states would move
twice across the width of the sample. Furthermore, the amplitude of the instantaneous velocity of the vortices would be
expected to vary continuously between maximum and minimum values as the relative proportion of the two dynamic
states changed. This periodic velocity response would be expected to lead to concomitant variations in the output voltage
amplitude, thereby accounting for the V 0 (t) oscillations.
The interpretation, as outlined above, is suggestive in that
could explain many features of the V 0 (t) oscillations such as
the relationship between t osc and t tr , the threshold values of
a t and I, and the fact that the oscillation period is independent of the modulation frequency. However, there are a number of important questions that remain to be answered, particularly in relation to the specific feedback mechanism
involved. One implication of the state boundary interpretation is that the dynamic state developing across the incoming
surface would always be different from that leaving the
sample at the outgoing surface: That is to say, where the
536
S. N. GORDEEV et al.
vortices leaving the sample were in an ordered dynamic
state, the state forming at the incoming surface would be
disordered and vice versa. It is not clear how such a situation
could develop but we tentatively suggest that it might have
something to do with ‘‘tensions’’ arising between the different dynamic states. If we accept that there is a difference
between the mobility of the two dynamic states, then we
would expect a two-state system to have a mobility that differed from that of a system consisting entirely of vortices in
one or other of these states. Thus, where there was a mobile
ordered state at the outgoing surface this might be expected
to ‘‘pull’’ the disordered state along with a higher velocity
than its ‘‘free-drift velocity.’’ The resulting tension between
dynamic states could perpetuate the formation of the disordered state at the incoming surface. In a similar fashion, the
presence of a disordered phase at the outgoing surface could
restrict the motion of the incoming vortices allowing them to
form into a well-ordered state. The qualitative arguments
outlined above give some idea as to how oscillations might
arise; however, more specific theoretical work will definitely
be required if rigorous explanation of these intriguing oscillatory effects is to be obtained.
Further insight into the mechanism of the oscillations
might also be provided by effects that occur in the quasi-onedimensional linear chain compounds such as NbSe3 or
K0.3MoO3. At a critical temperature, known as the Peierl’s
temperature, these systems undergo a transition into a
charge-density-wave ~CDW! state that manifests collectivemode dynamics.3 There are a number of similarities between
the dynamics of this collective mode and that of a driven flux
line lattice.34 In the dc driven state of a CDW system, sharp
peaks have been observed in the frequency spectrum of the
voltage response at a fundamental frequency and subsequent
harmonics.35 These spectra are similar in appearance to the
Fourier spectra extracted from our own data ~see Fig. 9!.
Moreover, it has been shown that the frequency of the fundamental peak in the CDW spectra is proportional to the
applied current.36 Although not immediately apparent, a
similar dependence is displayed by our own results. An amplitude asymmetry can be considered to be equivalent to a
pure ac modulation superimposed onto a dc drive. By definition, amplitude asymmetry a a 5I dc /I ac ; thus this asymmetry is proportional to the dc bias. We have found that the
fundamental frequency ( f osc) of the induced oscillations is
proportional to the applied asymmetry and it follows that
f osc}I dc . Thus, as for the CDW case the fundamental frequency of the oscillations is proportional to the ‘‘underlying’’ dc drive current. In the case of CDW’s the peaks in the
Fourier spectra are associated with a periodic superstructure
that has been directly observed in various diffraction
experiments.37 The similarities between the Fourier spectra
presented in this paper and those observed in the CDW case
provide further support for the idea that the V 0 (t) oscillation
effects could also be explained in terms of the motion of
some sort of a periodic superstructure.
V. SUMMARY
Investigations have been carried out on the dynamics of
the vortex system in YBa2Cu3O72d single crystals at tem-
PRB 58
peratures just below the thermodynamic melting temperature. We have found that the response of the system depends
not only on the magnitude of the driving force but also on
the form of the current modulation. Pure oscillatory ~double
polarity! driving currents were observed to result in much
higher average voltage responses and lower levels of voltage
noise than in the case of a unidirectional ~single polarity or
direct! drive. On the basis of the striking differences between
these responses we identified them with two distinct dynamic
states characterized by different degrees of relative disorder.
We have proposed an explanation for how this distinction
could arise, the essential idea being that in the case of a
unidirectional drive the vortex system is given little or no
opportunity to heal after its disorderly flow across the surface
of the sample, whereas in the case of the double-polarity
drive, the backwards and forwards motion leads to a gradual
ordering into a coherently moving dynamic state.
In order to investigate the effect of varying the net velocity of the vortex system while keeping the driving force constant, we devised a technique based upon the use of asymmetrically modulated driving currents with relatively low
period or amplitude asymmetries (;1%). The response of
the vortex system in these cases was found to be highly
unusual, reflecting a dynamic state unlike any that has been
identified to date. In response to an asymmetrically modulated square-wave current ~modulation frequency f m
568 Hz! the voltage response amplitude (V 0 ) executed slow
periodic oscillations ~typically f osc;0.01 Hz! over a large
min
range of V 0 values ~in some cases V max
was as large as
0 /V0
3!. Moreover, the frequency of the oscillations was found to
be entirely independent of the drive frequency except at the
very-low-frequency limit where the oscillations broke down
altogether. Similar oscillation effects were observed when a
sinusoidal or triangular wave form with a superimposed dc
offset was applied, thereby demonstrating that the square
wave form of the modulation is not, in itself, crucial in establishing oscillations. We have found that the oscillatory
effects were confined to a relatively limited window of parameter space with maximum and minimum currents, asymmetries, and temperatures beyond which oscillations were
not observed. Over the parameter region for which oscillatory effects were observed, they demonstrated one very important characteristic: The oscillation period was found to
approximately correspond to the time that it would take vortices to drift coherently twice across the width of the sample.
We have proposed an interpretation of the oscillatory effects based upon the idea of two distinct dynamic states, one
more ordered than other, coexisting within the sample. These
states can be likened to the ‘‘ordered’’ and ‘‘disordered’’
states that we have associated, respectively, with pure oscillatory and unidirectional driving current modulations. In the
framework of our interpretation, the oscillations can be explained in terms of the slow drift of the boundary between
the two dynamic states, with one period of the oscillations
corresponding to the amount of time taken for this boundary
to move twice across the width of the sample. In this way,
we are able to account for the observed factor-of-2 correspondence between the period of the oscillations and the
transit time for vortices drifting coherently across the width
of the sample.
PRB 58
OSCILLATORY DYNAMICS OF THE DRIVEN VORTEX . . .
ACKNOWLEDGMENTS
This project was supported by the EPSRC ~UK!. We are
grateful to the following people for useful discussions: E.
Andrei, I. Aranson, S. Bhattacharya, V. Geshkenbein, T.
*On leave from the Moscow Institute of Radioengineering, Electronics and Automation, 117454 Moscow, Russia.
1
C. Tang, K. Wiesenfeld, P. Bak, S. Coppersmith, and P. Littlewood, Phys. Rev. Lett. 58, 1161 ~1987!; P. B. Bak, C. Tang, and
K. Wiesenfeld, Phys. Rev. B 38, 364 ~1988!.
2
M. R. Sarkardei and R. L. Jacobs, Phys. Rev. E 51, 1929 ~1995!;
R. L. Jacobs and M. R. Sardarkei, in Physics of Sliding Friction,
edited by B. N. J. Persson and E. Tosatti ~Kluwer Academic,
Dordrecht, 1996!, p. 449.
3
G. Grüner and A. Zettl, Phys. Rep. 119, 117 ~1985!; G. Grüner,
Rev. Mod. Phys. 60, 1129 ~1988!.
4
E. Y. Andrei, G. Deville, D. C. Glattli, and F. I. B. Williams,
Phys. Rev. Lett. 60, 2765 ~1988!; Y. P. Li, T. Sajoto, L. W.
Engel, D. C. Tsui, and M. Shayegan, ibid. 67, 1630 ~1991!.
5
R. Seshardi and R. M. Westervelt, Phys. Rev. B 46, 5142 ~1992!;
46, 5150 ~1992!; Phys. Rev. Lett. 70, 234 ~1993!.
6
J. M. Carlson and J. S. Langer, Phys. Rev. Lett. 62, 2632 ~1989!;
Phys. Rev. A 40, 6470 ~1989!.
7
G. Blatter, M. Feigellman, D. Geshkenbein, A. I. Larkin, and V.
M. Vinokur, Rev. Mod. Phys. 66, 1125 ~1994!.
8
T. Giamarchi, P. Le Doussal, Phys. Rev. Lett. 75, 3372 ~1995!;
Phys. Rev. B 55, 6577 ~1997!; C. Reichhardt, C. J. Olson, and F.
Nori, Phys. Rev. Lett. 78, 2648 ~1997!.
9
H. J. Jensen, A. Brass, and A. J. Berlinsky, Phys. Rev. Lett. 60,
1676 ~1988!; M. C. Faleski, M. C. Marchetti, and A. A. Middleton, Phys. Rev. B 54, 12 427 ~1996!.
10
T. Giamarchi and P. Le Doussal, Phys. Rev. Lett. 76, 408 ~1996!;
in Spin Glasses and Random Fields, edited by A. P. Young
~World Scientific, Singapore, 1997!.
11
K. Moon, R. T. Scalettar, and G. T. Zimanyi, Phys. Rev. Lett. 77,
2778 ~1996!.
12
G. D’Anna, P. L. Gammel, H. Safar, G. B. Alers, D. J. Bishop, J.
Giapitzankis, and D. M. Ginsberg, Phys. Rev. Lett. 75, 3521
~1995!.
13
A. C. Shi and A. J. Berlinsky, Phys. Rev. Lett. 67, 1926 ~1991!;
A. E. Koshelev and V. M. Vinokur, ibid. 73, 3580 ~1994!; S.
Ryu, M. Hellerqvist, S. Doniach, A. Kapitulnik, and D. Stroud,
ibid. 77, 5114 ~1996!.
14
S. Bhattacharya and M. J. Higgins, Phys. Rev. Lett. 70, 2617
~1993!; Phys. Rev. B 49, 10 005 ~1994!; 52, 64 ~1995!; A. C.
Marley, M. J. Higgins, and S. Bhattacharya, Phys. Rev. Lett. 74,
3029 ~1995!; M. J. Higgins and S. Bhattacharya, Physica C 257,
232 ~1996!.
15
U. Yaron, P. L. Gammel, D. A. Huse, R. N. Kleiman, C. S.
Oglesby, E. Bucher, B. Batlogg, D. J. Bishop, K. Mortensen,
and K. N. Clausen, Nature ~London! 376, 753 ~1995!.
16
S. Spencer and H. J. Jensen, Phys. Rev. B 55, 8473 ~1997!.
17
S. N. Gordeev, P. A. J. de Groot, M. Oussena, A. V. Volkozub, S.
Pinfold, R. M. Langan, R. Gagnon, and L. Taillefer, in Proceedings of the 8th International Workshop on Critical Currents in
Superconductors, edited by T. Matsushita and K. Yamafuji
~World Scientific, Singapore, 1996!, p. 211.
18
S. N. Gordeev, P. A. J. de Groot, M. Oussena, A. V. Volkozub, S.
537
Giamarchi, M. Indenbom, R. Jacobs, P. Kes, A. Koshelev,
W. Kwok, M. Oussena, V. Vinokur, V. Vlasko-Vlasov, and
A. Zhukov. R.G. and L.T. acknowledge funding from
NSERC ~Canada!, FCAR of Quebec, and CIAR ~Canada!.
L.T. acknowledges the support of the Sloan Foundation.
Pinfold, R. Langan, R. Gagnon, and L. Taillefer, Nature ~London! 385, 324 ~1997!.
19
S. N. Gordeev, P. A. J. de Groot, M. Oussena, A. V. Volkozub, S.
Pinfold, R. M. Langan, R. Gagnon, and L. Taillefer, J. Low
Temp. Phys. 105, 1147 ~1996!; S. N. Gordeev, P. A. J. de Groot,
M. Oussena, A. V. Volkozub, R. M. Langan, A. P. Rassau, R.
Gagnon, and L. Taillefer, Physica C 282-287, 2033 ~1997!; S. N.
Gordeev, A. P. Rassau, P. A. J. de Groot, R. Gagnon, and L.
Taillefer, in Applied Superconductivity, edited by H. Rogala and
D. H. A. Blank ~Institute of Physics, Bristol, 1997!, p. 37.
20
W. K. Kwok, G. W. Crabtree, J. A. Fendrich, and L. M. Paulius,
Physica C 293, 111 ~1997!.
21
R. Gagnon, C. Lupien, and L. Taillefer, Phys. Rev. B 50, 3458
~1994!.
22
S. Fleshler, W. K. Kwok, U. Welp, V. M. Vinokur, M. K. Smith,
J. Downey, and G. W. Crabtree, Phys. Rev. B 47, 14 448 ~1993!.
23
H. Safar, P. L. Gammel, D. A. Huse, D. J. Bishop, J. P. Rice, and
D. M. Ginsberg, Phys. Rev. Lett. 69, 824 ~1992!; W. K. Kwok,
S. Fleshler, U. Welp, V. M. Vinokur, J. Downey, G. W. Crabtree, and M. M. Miller, ibid. 69, 3370 ~1992!.
24
W. K. Kwok, J. A. Fendrich, C. J. van der Beek, and G. W.
Crabtree, Phys. Rev. Lett. 73, 2614 ~1994!; H. Safar, P. L. Gammel, D. A. Huse, G. B. Alers, D. J. Bishop, W. C. Lee, J. Giapintzakis, and D. M. Ginsberg, Phys. Rev. B 52, 6211 ~1995!.
25
A. I. Larkin, M. C. Marchetti, and V. M. Vinokur, Phys. Rev.
Lett. 75, 2992 ~1995!.
26
E. Zeldov, J. R. Clem, M. McElfresh, and M. Darwin, Phys. Rev.
B 49, 9802 ~1994!; M. Däumling and D. C. Larbalestier, ibid.
40, 9350 ~1989!.
27
I. Aranson and V. M. Vinokur, Phys. Rev. Lett. 77, 3208 ~1996!;
Phys. Rev. B 57, 3073 ~1998!.
28
D. W. Braun, G. W. Crabtree, H. G. Kaper, A. E. Koshelev, G. K.
Leaf, D. M. Levine, and V. M. Vinokur, Phys. Rev. Lett. 76,
831 ~1996!.
29
F. Pardo, F. de la Cruz, P. L. Gammel, C. S. Oglesby, E. Bucher,
B. Batlogg, and D. J. Bishop, Phys. Rev. Lett. 78, 4633 ~1997!.
30
W. Henderson, E. Y. Andrei, M. J. Higgins, and S. Bhattacharya,
Phys. Rev. Lett. 77, 2077 ~1996!.
31
V. M. Vinokur and T. Nattermann, Phys. Rev. Lett. 79, 3471
~1997!.
32
N. Gro” nbech-Jensen, A. R. Bishop, and D. Dominguez, Phys.
Rev. Lett. 76, 2985 ~1996!.
33
S. Scheidl and V. M. Vinokur, Phys. Rev. B 56, 8522 ~1997!.
34
S. N. Coppersmith and A. J. Millis, Phys. Rev. B 44, 7799
~1991!.
35
R. M. Fleming and C. C. Grimes, Phys. Rev. Lett. 42, 1423
~1979!.
36
P. Monceau, J. Richard, and M. Renard, Phys. Rev. B 25, 931
~1982!; A. Zettl and G. Grüner, ibid. 29, 755 ~1984!; S. E.
Brown and G. Grüner, ibid. 31, 8302 ~1985!.
37
Charge Density Waves in Solids, edited by L. P. Gor’kov and G.
Grüner ~Elsevier, Amsterdam, 1989!.