PHYSICAL REVIEW B, VOLUME 65, 094507
Vortex-state metastability at low-field disorder-order transition
G. Ravikumar,1,2 H. Küpfer,1 A. Will,1 R. Meier-Hirmer,1 and Th. Wolf3
1
Institut für Technische Physik, Forschungszentrum Karlsruhe GmbH and Universität Karlsruhe, D-76021 Karlsruhe, Germany
2
Technical Physics and Prototype Engineering Division, Bhabha Atomic Research Centre, Mumbai-400085, India
3
Institut für Festkörperphysik, Forschungszentrum Karlsruhe GmbH, D-76021 Karlsruhe, Germany
共Received 2 August 2001; published 5 February 2002兲
We report history-dependent critical currents across the disordered solid to Bragg glass transition occurring
in the low-field region in V3 Si and NdBa2 Cu3 O7⫺x single crystals. This is in addition to the history dependence usually observed near the high-field Bragg-glass–disordered-solid transition. It is further shown that the
metastable frozen-in disordered vortex matter coexists with the ordered phase in the entire Bragg glass region.
We conclude that the amount of the metastable disordered phase prior to the high-field Bragg-glass–
disordered-solid transition is crucial in nucleating further growth of the disordered phase near the transition.
This mechanism results in the onset of the high-field peak to occur at lower fields when the frozen-in disordered phase prior to the transition is larger.
DOI: 10.1103/PhysRevB.65.094507
PACS number共s兲: 74.60.Ge, 74.72.Jt
In the mixed state of type-II superconductors, just above
the first critical field, the intervortex distance is comparable
to the London penetration depth. In this low-field regime,
pinning is always dominant and a highly disordered flux line
lattice is therefore preferred. As the field is progressively
increased, a growing intervortex interaction stabilizes the
quasiordered Bragg glass phase 共Fig. 1兲. The field where this
low-field disorder-order 共LF-O/D兲 transition1,2 occurs depends on the extent of quenched disorder in the superconductor. At further higher fields, Bragg glass phase again undergoes a transition to a highly disordered state.3 This highfield order-disorder 共HF-O/D兲 transition 共Fig. 1兲 is
characterized by a sharp increase in critical current density
J c , known as peak effect or second magnetization peak.4,5
The HF-O/D transition is usually accompanied by vortexstate metastability and history dependence in J c . 6 –18 In addition, we present experimental results to show the history
dependence in J c also near the LF-O/D transition. It was
recently argued that both the LF-O/D and HF-O/D transitions
are first order in nature,2 consistent with observed history
effects. There is indeed a growing consensus that it is a
quenched-disorder-induced first-order transition of vortex
matter.2,15,19,20
The physical mechanism for the metastability in the vicinity of the HF-O/D transition is a subject of current discussion. Paltiel et al.2,23 proposed that the metastable-disordered
phase is injected at surface imperfections which dynamically
coexist with the stable-ordered phase prior to the transition.
Based on our experiments, we shall argue that the presence
of a small amount of metastable-disordered phase prior to the
transition nucleates the growth of the disordered phase by
accumulating more vortices as the field is increased. From
our experiments, it appears that this mechanism is predominant over that proposed by Paltiel et al.2,23
Magnetization experiments on a V3 Si crystal (1.6 mm
⫻0.5 mm⫻0.3 mm) with weak pinning (T c ⫽17 K) and a
twin-free NdBa2 Cu3 O7⫺x (Nd123) crystal (1.4 mm
⫻0.5 mm⫻0.2 mm) with T c ⫽95 K 共Ref. 18兲 are carried
out using an Oxford vibrating sample magnetometer. Nd123
crystal was oriented with its c axis parallel to the field. V3 Si
crystal was oriented with the field parallel to its smallest
0163-1829/2002/65共9兲/094507共4兲/$20.00
dimension. Figure 2 shows the magnetic moment m versus
field H in increasing 共forward curve兲 and decreasing 共reverse
curve兲 fields for the V3 Si sample at 9.5 K 共detailed measurements are carried out at this temperature although similar
results can be obtained in a wide temperature range兲 measured with a field sweep rate of 1.2 T/min. Forward curve is
measured from a sufficiently large negative field so that induced currents are unidirectional. The initial sharp fall in the
magnetization hysteresis 共or J c ) is associated with the disordered solid to Bragg glass transition as discussed above. We
identify the HF-O/D transition by the onset (H on ) of a sharp
decrease in magnetic moment occurring on the forward curve
at around 6 T.5 Such a choice is justified later. The exact
location of the transition is, however, controversial due to the
occurrence of metastability.
It is common to investigate the history dependent J c by
means of minor magnetization curves.9–11,13,14,17,18 For instance, near the HF-O/D transition, the minor curves starting
on the forward curve 共point A in Fig. 2兲 saturate and do not
meet the reverse curve until much lower fields. On the other
hand, minor curves starting on the reverse curve 共point B in
Fig. 2兲 overshoot the forward curve. As discussed in Ref. 11,
FIG. 1. Schematic phase diagram in the mixed state of a type-II
superconductor depicting the Bragg glass and disordered solid
phases. Thermal effects that are important in high-T c materials are,
however, ignored. H c1 and H c2 are the first and second critical
fields respectively. HF-O/D and LF-O/D lines define the high- and
low-field boundaries of the Bragg glass phase 关see Paltiel et al.
共Ref. 2兲兴.
65 094507-1
©2002 The American Physical Society
RAVIKUMAR, KÜPFER, WILL, MEIER-HIRMER, AND WOLF
PHYSICAL REVIEW B 65 094507
FIG. 2. Forward and reverse m-H curves in a V3 Si sample
indicating the HF-O/D and LF-O/D transitions and the peak field
H p . Typical behavior of minor curves starting from the forward
共solid line兲 and reverse curves 共dotted line兲 near the HF-O/D transition. Close to H p , the minor curves readily merge into the main
loop.
this suggests that J c is larger on the decreasing field branch
than on the increasing field branch. However, close to the
peak position H p , the minor curves 共starting at C and D兲
merge into the main magnetization loop, indicating that J c is
independent of magnetic history.
In Figs. 3 and 4, we present minor curves in the low-field
region 共around 1 T兲 for V3 Si and Nd123 samples, respectively. Here, the minor curves starting from the forward
curve overshoot the reverse curve 关Figs. 3共a兲 and 4共a兲兴. On
the other hand, the minor curves starting from the reverse
curve saturate and do not meet the forward curve until a
much higher field 关Figs. 3共b兲 and 4共b兲兴. This behavior, exactly opposite to that seen near the HF-O/D transition 共Fig.
2兲, is observed over a wide temperature range, for fields
above 0.2 T. This behavior of minor curves clearly shows the
history-dependent J c near the LF-O/D transition; i.e., J c is
larger on the increasing field branch than on the decreasing
field branch. This conclusion is just opposite to that near the
HF-O/D transition. As expected, the effect in Nd123 is much
weaker than in V3 Si, because stronger thermal fluctuations in
high-T c materials significantly weaken the observable metastability.
The history dependence in J c near the HF-O/D transition
was recently explained by invoking supercooling9,11,15,21,22 of
the high-field disordered solid phase 共high J c ) upon lowering
the field below the HF-O/D transition line 共path 1 in Fig. 1兲.
Similarly, the quasiordered Bragg glass phase is
superheated15,22 by increasing the field above this line.14
Consequently, vortex matter is relatively more ordered
共lower J c ) on the forward curve than on the reverse curve.
The terms supercooling and superheating are usually associated with first-order transitions where the free energy has
two minima. In the case of the order-disorder transition multiple free energy minima and metastable vortex configurations due to quenched disorder may effectively produce supercooling and superheating effects.
The history-dependent J c in the low-field region can be
understood in a similar way. Here, the low-field disordered
solid phase is superheated upon increasing the field 共forward
curve兲 above the LF-O/D line 共path 2 in Fig. 1兲. On the other
FIG. 3. 共a兲 Minor curves starting from the forward curve in the
low-field region overshoot the reverse curve. 共b兲 In the low-field
region, minor curves starting from the reverse curve saturate and do
not meet the forward curve until a field of about 7 T (⬎H on ). Note
the shift in the HF-O/D transition (H on ) to higher fields on the
minor curves. The inset shows H on plotted as a function of the
absolute value of m at a field of 5.5 T, on different minor curves.
hand, upon decreasing the field below this line, the Bragg
glass phase is supercooled, resulting in a relatively more ordered 共thus low J c ) vortex lattice on the reverse curve than
on the forward curve.
J c ceases to be history dependent close to the peak field
H p , beyond which only the disordered phase can exist and
the ordered phase cannot be present even in the metastable
form. Similar arguments apply on the low-field side below a
certain field H L 共about 0.2 T兲 where J c is no longer history
dependent. The location of both HF-O/D and LF-O/D transitions is limited by these two fields H p and H L , respectively. The low-field minor curves can be used to determine
the field H L , below which the vortex matter is unambiguously in a disordered equilibrium state.
We now refocus on the minor curves drawn in Fig. 3共b兲,
which do not meet the forward curve up to a field of about 7
T, which is well above H on . Below H on , they are almost
parallel, each corresponding to a different metastable state
with a specific amount of frozen-in disordered phase. Clearly,
the position of the onset seems to be very sensitive to this
frozen-in disorder. In the main panel, we show two dotted
lines, which are linear extrapolations of the m-H data 共i兲
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VORTEX-STATE METASTABILITY AT LOW-FIELD . . .
PHYSICAL REVIEW B 65 094507
FIG. 4. Low-field 共below 1 T兲 minor curves in Nd123 crystal at
69 K. In 共a兲 minor curves starting from the forward curve 共not
shown兲 overshoot the reverse curve as in Fig. 3共a兲. 共b兲 Minor curves
共dotted lines兲 starting from the reverse curve 共not shown兲 do not
meet the forward curve as in Fig. 3共b兲.
above 5 T on the low-field side and 共ii兲 below 7 T on the
high-field side. The intersection point is used as the criterion
to determine H on , the position of HF-O/D transition, on different minor curves. In the inset, H on measured on different
minor curves is plotted as a function of the m value at 5.5 T
on that minor curve. As shown in the inset, this criterion,
used in obtaining H on , gives a total shift in the transition of
about 0.2 T. Alternative criteria—for instance, the field
where the m value changes significantly from an almost constant value below the transition—would give a much higher
spread 共about 0.6 T兲 in the transition field. Assuming that the
different magnetic moments at 5.5 T are a measure of the
different extents of frozen-in metastable disorder prior to the
HF-O/D transition, we conclude that the position of the transition shifts to lower fields with increasing amount of
frozen-in metastable disorder. Within the mechanism proposed by Paltiel et al.,23 it is hard to imagine how the starting
field of the minor curves 共for instance at 0.5 and 1.0 T兲 can
affect the disordered phase present at 5.5 T, because, on all
the minor curves, new vortices entering the system experience identical conditions at the edges. We propose the following mechanism to understand the shift in the transition
field on different minor curves.
Close to H on , apparently a small amount of metastable
disordered phase can nucleate further growth of the disordered phase by accumulating more vortices as the field is
increased. The growth is faster when the initial frozen-in
disorder is larger 共or more nucleating sites兲. In the absence of
frozen-in disorder, vortex matter continues to be relatively
ordered up to much higher fields, leading to an apparent shift
in the onset field to higher values. On the minor curves, the
frozen-in metastable disordered phase prior to the transition
is smaller and therefore the onset of the peak occurs at higher
fields. In other words, the frozen-in disordered phase 共nucleating sites兲 prevents superheating of the ordered phase by
triggering the transition, analogous to the case in standard
FIG. 5. 共a兲 Minor hysteresis loops initiated on the forward
curve, obtained 共around 5.15 T兲 by repeatedly cycling the field,
collapse into the loop shown as solid squares after many cycles. 共b兲
A similar experiment near 4.05 T. Note that the saturated loop is
now closer to the reverse curve, while it is closer to the forward
curve at 5.15 T. The main hysteresis loop 共forward and reverse
curves兲 in the respective field ranges is in the dark line.
first-order transitions. We may therefore argue that H on measured on the forward curve, with a higher amount of
frozen-in disorder prior to the transition, is likely to be closer
to the real HF-O/D transition.
In the following experiment, we demonstrate the possibility of frozen-in metastable disorder, in the entire Bragg glass
region 共between LF-O/D and HF-O/D lines兲, both on increasing and decreasing field branches. It was shown earlier
that the metastable-disordered phase can be annealed away
or fully eliminated by repeatedly cycling the field by a small
amplitude.14,15 The idea is, when the field is cycled, vortices
move from their metastable configuration and reorganize into
a stable configuration 共so-called process of annealing兲.
Figure 5共a兲 shows a part of the main hysteresis loop
around 5.15 T where magnetization hysteresis is significantly
lower than in the peak region as well as in the low-field
region 共see Fig. 2兲. We also plot the minor hysteresis loops
starting from the forward curve, obtained by repeatedly cycling the field with an amplitude ⌬H⫽0.1 T, which is more
than adequate to reverse the induced currents throughout the
sample. As shown in Fig. 5共a兲, the width of the minor hysteresis loop collapses with each successive field cycle and
saturates on the loop shown by connected squares. The final
width of the loop after several cycles 共for clarity some of the
loops are omitted兲 is almost an order of magnitude smaller
than the width of the main hysteresis loop at that field. The
saturated hysteresis loop obtained is essentially the same
even when the minor loops are initiated from the reverse
094507-3
RAVIKUMAR, KÜPFER, WILL, MEIER-HIRMER, AND WOLF
PHYSICAL REVIEW B 65 094507
curve, indicating the existence of a unique stable vortex
state.14,15
We show a similar result near 4.05 T in Fig. 5共b兲, i.e.,
minor loops starting from the forward curve with repeated
field cycling, collapsing to a stable state 共again some of the
intermediate loops are omitted for clarity兲. We note that the
final saturated loop 共shown as solid squares兲 at 4.05 T is now
closer to the reverse curve while at 5.15 T it is closer to the
forward curve. This is expected because at 4.05 T, the supercooled metastable phase from above the HF-O/D line 共reverse curve兲 is annealed better than at 5.15 T. On the other
hand, the superheated disordered phase from the LF-O/D line
共forward curve兲 is more annealed at 5.15 T than at 4.05 T. In
conclusion, the stable vortex state in the Bragg glass region
is much more ordered than it appears from the width of the
main hysteresis loop.
The extent of collapse in the magnetization hysteresis
upon field cycling suggests a significant presence of the
metastable disordered phase coexisting with the thermodynamically stable Bragg glass phase, both on forward and
reverse curves. On the forward curve, the low-field disordered phase 共stable below the LF-O/D line兲 extends deep
into the Bragg glass region as a superheated metastable
phase. Similarly, the disordered phase, stable above the HFO/D line, extends deep into the Bragg glass region as a supercooled metastable phase. We further note that upon in-
creasing the field after eliminating the metastable phase by
repeated field cycling, H on is much higher than that on the
forward curve. This further supports our point that the
amount of frozen-in disorder is crucial in determining the
onset of the peak effect.
In conclusion, we reported the occurrence of historydependent critical currents near the low-field disorderedsolid–Bragg-glass transition in V3 Si and NdBa2 Cu3 O7⫺x
single crystals. The nature of the history dependence in critical currents near this transition region is just the opposite to
that seen near the high-field Bragg-glass–disordered-solid
transition. It is also shown that the disordered vortex phase
predominantly coexists as a metastable phase in the Bragg
glass region with the stable-ordered phase. The onset of the
peak effect shifts to lower fields with increasing amount of
metastable disordered phase. We conclude that the presence
of the disordered phase prior to the Bragg-glass–disorderedsolid transition is crucial in nucleating the growth of the
disordered phase at the onset of the peak effect. We further
argue that the onset field may be closer to the true position of
the HF-O/D transition.
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