Vortex dynamics differences in YBaCuO at low temperatures for H||ab planes due to twin boundary pinning anysotropy

Abstract

We measured magnetization, M, of a twin-aligned single crystal of YBa2Cu 3Ox
(YBaCuO), with Tc = 91 K, as a function of temperature, T, and magnetic field,
H, with H applied along the ab planes. Isothermal M-vs.-H and M-vs.-time curves
were obtained with H applied parallel and perpendicular to the twin boundary,
TB, direction. M-vs.-H curves exhibited two minimums below 38 K, which
resembled similar curves that have been obtained in YBaCuO for H parallel to c
axis. Above 12 K, the field positions of the minimums for H||TB and HperpTB
were quite similar. Below 12 K, the position of the second minimum, Hmin,
occurred at a higher field value with H||TB. Below 6 K, only one minimum
appeared for both field directions. At low temperatures, these minimums in the
M-vs.-H curves produced maximums in the critical current. It was determined
that vortex lines were expelled more easily for H||TB than for HperpTB, and,
therefore, below a certain field value, that Jc(HperpTB) was larger than
Jc(H||TB) . At T<12 K with H||TB, the relaxation rate for flux lines leaving
the crystal was found to be different from that for flux entering the crystal.
We also observed flux jumps at low temperatures, with their sizes depending on
the orientation of magnetic field with respect to the TBs.

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arXiv:cond-mat/0412331v1 [cond-mat.supr-con] 13 Dec 2004
Vortex dynamics differences in YBaCuO at low temperatures for H||ab planes due to
twin boundary pinning anysotropy.
S. Salem-Sugui, Jr.1, A. D. Alvarenga2, M. Friesen3, K. C. Goretta4, O. F. Schilling5, F. G. Gandra6, B. W. Veal7,
P. Paulikas7
1Instituto de F´ ısica, Universidade Federal do Rio de Janeiro
C.P.68528, 21945-970 Rio de Janeiro, RJ, Brasil
2Centro Brasileiro de Pesquisas Fisicas, Rua Dr. Xavier Sigaud,150,
22290-180 Rio de Janeiro, RJ, Brazil
3Department of Materials Science and Engineering, 1500 Engineering
Drive, University of Wisconsin, Madison, Wisconsin 53706
4Energy Technology Laboratory, Argonne National laboratory, Argonne,
Illinois
60439-4838
5Departamento de F´isica, Universidade Federal de Santa Catarina,
88040-900 Florian´ opolis,SC, Brazil.
6Instituto de Fisica, UNICAMP, CP6185, 13083-970 Campinas, SP, Brazil
7Materials Sciences Division, Argonne National Laboratory, Argonne,
Illinois, 60439-4838
We measured magnetization, M, of a twin-aligned single crystal of YBa2Cu3Ox (YBaCuO), with
Tc = 91 K, as a function of temperature, T, and magnetic field, H, with H applied along the ab
planes. Isothermal M-vs.-H and M-vs.-time curves were obtained with H applied parallel (?) and
perpendicular (⊥) to the twin boundary, TB, direction. M-vs.-H curves exhibited two minimums
below 38 K, which resembled similar curves that have been obtained in YBaCuO for H?c axis.
Above 12 K, the field positions of the minimums for H?TB and H⊥TB were quite similar. Below 12
K, the position of the second minimum, Hmin, occurred at a higher field value with H?TB. Below
6 K, only one minimum appeared for both field directions. At low temperatures, these minimums
in the M-vs.-H curves produced maximums in the critical current. It was determined that vortex
lines were expelled more easily for H?TB than for H⊥TB, and, therefore, below a certain field value,
that Jc(H⊥TB) was larger than Jc(H?TB) . At T <12 K with H?TB, the relaxation rate for flux
lines leaving the crystal was found to be different from that for flux entering the crystal. We also
observed flux jumps at low temperatures, with their sizes depending on the orientation of magnetic
field with respect to the TBs.
Introduction
Twin planes are ubiquitous in the high-temperature superconductor YBa2Cu3Ox(YBaCuO). Under microscopic
analysis they appear as flat, slab-like domains of micrometer thickness. The domains are coherent and are oriented in
various [110] directions. The a and b axes are inverted in neighboring domains. The domain boundaries are commonly
referred to as twin boundaries (TBs). The boundary region has a structure different from that of the bulk crystal,
including a relative deficiency of oxygen [1–3] or excess of impurities that can accumulate during crystal growth. TBs
form strong vortex pinning centers and are responsible for a rich variety of transport anisotropies. Under typical
growth conditions, these anisotropies cannot be observed readily because the neighboring TBs are not aligned. The
gross vortex dynamics of samples with such TBs are determined by the sample’s texture. Twin-aligned YBaCuO
samples can, however, be formed, and such samples provide windows into the effects on superconducting properties
of TBs.
Initial studies of twin-aligned YBaCuO single crystals revealed strong anisotropies for magnetic fields perpendicular
to the c axis of the crystal, H⊥c, for TBs oriented either parallel or perpendicular to the field [3–6]. These studies
included isothermal measurements of resistivity and magnetic hysteresis. Additional angular dependencies between
H and TBs were reported in [ [7]] and [ [8]]. The effects of TBs on vortex dynamics were studied by magneto-optical
measurements for H parallel and perpendicular to the c axis [9–11]. More recent studies of twin-aligned single crystals
of YBaCuO include transport measurements as a function of current in the ab plane [12] and Bitter decoration
experiments under tilted magnetic fields [13]. Vortex-pinning [14] and flux-creep measurements have been studied
through use of ac probes for the configurations H parallel to the TBs, H?TB, and H perpendicular to the TBs, H⊥TB
[15].
Previous TB-based studies of YBaCuO focused mainly on higher temperatures, which motivated the present work
in the temperature region 50 K <T <2 K. We studied the effects of TB pinning anisotropy on magnetic-hysteresis
and magnetic-relaxation curves for a twin-aligned single crystal of YBaCuO, with the magnetic field applied in the ab
plane for two directions, namely H||TB and H⊥TB. The study revealed interesting features in magnetic hysteresis and
magnetic relaxation curves due to TB vortex-pinning anisotropy, that, to our knowledge, have not yet been reported:
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We observe the existence of a second minimum in magnetic hysteresis curves below 38 K, which temperature behavior
below 12 K for H||TB is quite different than the one observed when H ⊥ TB. The temperature behavior (below
12 K) of the relaxation rate for H||TB (studied for H = 3 T), is also quite distinct of the behavior observed when
H ⊥ TB. At low temperatures, maximums in the critical current Jcoccurred at the positions of the minimums in
the M(H) curves. We also observe flux-jumps in the magnetization curves at low temperatures, with sizes depending
wether H is applied parallel or perpendicular to TB.
Flux jumps result from thermomagnetic instabilities associated with dissipative heating (either flux flow or
avalanche). If the dissipative heating cannot diffuse through the sample, it can increase the local temperature,
possibly even above the critical temperature, Tc, producing a jump in magnetization. Such jumps may occur in
response to changes in the external field if the magnetic-diffusion time is shorter than the thermal-diffusion time. The
size of a jump depends on the rate of magnetic field increase, dH/dt. Theoretically, the stability criterion [16,17]
defines a critical thickness below which flux jumps do not occur. Experimentally, such results have been confirmed for
melt-textured YBaCuO with the magnetic field applied perpendicular to the c axis, H⊥c [18]. Additional asymmetries
have been observed in the different branches of magnetization hysteresis curves, M(H), for melt-textured YBaCuO at
T <6 K, with a greater prevalence of flux jumps occurring with increasing field than with decreasing field [18]. These
results are in contrast to those for H||c, for which no hysteresis asymmetries have been observed [18,19]. For practical
applications, especially those involving high currents, characterization and management of flux jumps become critical.
We therefore analyzed the observed flux jumps in some detail.
Experimental details
The sample was a single crystal of YBaCuO with Tc= 91 K and dimensions ∼1 x 1 x 0.1 mm. Figure 1 shows
an enlarged (100x) photo of the sample’s surface. All twin boundaries within the crystal were parallel and extended
across the entire thickness of the sample, as confirmed by microscopic analysis. The density of TBs was estimated to
be 45 twins per mm. The sample had an approximately square shape, with one slightly rounded corner, and the twin
boundaries displaced perpendicular to the larger diagonal.
Magnetization and magnetic-relaxation data were taken after cooling the sample in zero applied magnetic field. A
commercial magnetometer (Quantum Design PPMS-9T) was utilized for the measurements. The magnetic signal of
the sample (plus sample holder) was obtained from the inductive signal of a pick-up coil, which appeared because
of motion of the sample through the coil in a homogeneous magnetic field, which we term a scan. Each set of
magnetization data represents the average of three scans. Magnetization-vs.-field, M(H), curves were obtained at
fixed temperatures ranging from 2 to 50 K. For a fixed applied field of H = 3 T, magnetic-relaxation measurements
were obtained at 60 s intervals over a period of 3600 s, for both the upper and lower branches of a hysteresis curve.
The remanent magnetization at zero field was also recorded. The value of 3 T was chosen to minimize the effects of
field penetration during the measurements [20]. We refer to the increasing field magnetization as Min, the decreasing
field magnetization as Mout, and the remanent magnetization as Mrem. The latter two signals were obtained after
first increasing the field to 9 T.
In the temperature range 2-12 K, all relaxation measurements were obtained for both field orientations with respect
to the TBs. Care was taken to assure that the magnetic field was applied in the ab plane. The sample was mounted
on a flat surface machined into the center of a 3 cm wooden cylinder that fit snugly into a straw that was inserted into
the magnetometer. An optical microscope with polarized light was used for sample mounting. The angle between the
TBs and the magnetic field H was estimated to be accurate to <2◦. After the experiment was concluded, we measured
M(H) for the sample holder at all relevant experimental temperatures to account for background corrections.
Results and discussion
Figure 2 contains selected M-vs.-H curves obtained for both orientations of H with respect to the TBs at T =
25, 15, and 8 K. The arrows in Fig. 2 are pointed at the curves obtained at 8 K. From left to right in the figure,
they are: the first arrow is the first minimum Hpen, which is associated to field penetration; the second arrow
is the local maximum Hon, which is associated with the field at which pinning sets in; and the third and forth
arrows are the second minimums Hmin, which are reminiscent of the second magnetization peaks observed in high-
Tcsuperconductors for H||c axis. The decreasing-field portions of M-vs.-H curves do not show any maximum. The
third and forth arrows show, respectively, Hmin for H⊥TB and Hmin for H||TB. These three fields–Hpen, Hon, and
Hmin–were clearly present in all M-vs.-H curves obtained at 8-33 K.
M-vs.-H curves obtained at or above 38 K exhibited only a single minimum, which is associated with Hpen. M-
vs.-H curves obtained below 8 K also exhibited only one minimum, which, in this particular case, may possibly be
associated with Hmin. This conjecture will be discussed below.
The positions of Hpen, Hon, and Hmin for both directions of applied field were approximately the same for
temperatures 12 K < T < 38 K, but a change in the field position of Hmin was noted below 12 K. Such a change
in field position can be observed in the M(H) curves at 8 K depicted in Fig.2, in which Hmin for H||TB occurred
at a much higher field than did Hmin for H⊥TB. The physical reason for the shifting of Hmin, as shown at 8 K, to
occur only below 12 K is not clear. It may be related to the temperature behavior of the TB barriers found below
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12 K, as will be discussed below. The insets in Fig.2 show the angular dependence at 8 K of M(H) for small angles.
These data will also be discussed below.
Figure 3 shows M(H) curves obtained at 2 K in the main figure and at 4 K in the inset. These curves clearly show
the existence of a single minimum. As observed for Hmin at 8 K, the field position of the minimum is higher for
H||TB than for H⊥TB, which suggests that the minimum observed below 8 K is associated with Hmin rather than
with Hpen. The inflection points apparent in the curves of Fig. 3 for fields H < Hmin, which are most visible in
the H⊥TB curves, are possibly related to Hon. Flux jumps are also evident for H||TB. A smaller flux jump was also
observed at 4 K (inset of Fig. 3) for H||TB only.
The principal differences, that are due to TBs, among the curves in Fig. 2 and between those in Fig 3 at fixed
temperatures are as follows.
(1) The values of ∆M for intermediate and higher fields were higher for H||TB than for H⊥TB. This result been
obtained before; it is due to twin-boundary pinning [3].
(2) The diamagnetic signal for H < Hpen was higher for H||TB than for H⊥TB. This result has been also observed
previously [3]. For the same applied field value H, differences in the diamagnetic signal in the field-penetration region,
just above Hc1, suggest that the local field Hiat the sample surface is smaller when H⊥TB than when H||TB. Such a
result is possible if the demagnetization factor of the sample is higher for the configuration H⊥TB: Hi= H- NM, where
N is the demagnetization factor. Our sample was a thin slab and the experiment was conducted with the magnetic
field lying in the plane of the slab. This configuration suggests that demagnetization fields were quite small. On the
other hand, although the crystal’s face was approximately square, one corner was rounded, with the twin boundaries
displaced perpendicular to the larger diagonal. Therefore, the sample geometry for H||TB was significantly different
than for H⊥TB, and the demagnetization factor for each case would be expected to be different, which may explain
the differences in the diamagnetic signal observed for each case.
(3) After decreasing the field until H = 0 (the decreasing-field branches of the M-vs.-H curves), the remanent
magnetization defined as M(H = 0) was higher for H⊥TB than for H||TB [3]. In fact, magnetization in the decreasing-
field branch started to become higher for H⊥TB below a certain field, the value of which increased as temperature
decreased. This response was observed for all M(H) curves and can be clearly observed in Fig. 4, in which the
estimated critical current density, Jc, vs. field is plotted for temperatures below 10 K, for both field directions with
respect to the TBs. Jc in A/cm2 was estimated from the Bean critical-state model [21]. Below a certain field, Jc
for H⊥TB was always higher than Jcfor H||TB; this response was observed for all curves from 50 K to 2 K. This
relationship between Jcand the TBs does not appear to have been discussed in the literature. It is interesting to note
that as temperature was lowered, at a field of a few Tesla a broad maximum in Jcemerged (maximums were clear
at 4 K). The position of each maximum in Jcappeared to be related to the respective field position of the minimum
(Hmin) in each M(H) curve at 4 K.
After obtaining the data set at 2-50 K, we measured M(H) curves at 8 K, with the applied magnetic field tilted
within the plane by a small angle (5◦<θ <10◦) relative to the original directions, H⊥TB (θ = 90◦) and H||TB (θ
= 0◦). (The magnetic field remained in the ab plane when the sample was rotated.) Although a full set of angular-
dependence measurements was beyond the scope of this work, this limited set of measurements allowed us to check
for possible edge (or geometric) effects in the M(H) curves at temperatures at which two minimums had been clearly
resolved. The results for each set of M(H) curves are shown in the insets of Fig. 2. Inspection of the insets reveals
two facts:
(a) Rotation of the TBs by a small angle with respect to the applied field changed the position of the second
minimum for H||TB (Hmin was displaced by a small field value after the sample was rotated), but not for H⊥TB.
Change of the position of Hmin only for the case H||TB suggests that Hmin is related to TB pinning anisotropy.
We further speculate that the absence of the corresponding second peak in the decreasing-field branch of the curve
may be due to the fact that with decreasing of the field, flux lines could leave the sample relatively easily through the
TBs.
(b) Rotation of TBs with respect to the applied field by a small angle had considerable effect on the shape of the
M(H) curves in the field-penetration region (Hpen <H< Hon) for both H⊥TB and H||TB, which provides evidence
for the importance of edge (or geometric) effects in this field region. The M(H) curve for θ = 7◦(left inset of Fig. 2) is
apparently rotated with respect to the M(H) curve for θ = 0◦in the same figure. This apparent rotation in the M(H)
curve is simply due to a change in the position of the sample in the holder: After rotating the sample, it was displaced
out of the middle of the its holder, and the signal due to the sample holder (i.e., the background magnetization) was
not subtracted correctly, which produced the apparent rotation. In our experimental set-up, the sample was fixed to
the sample holder with GE varnish. Changing the sample’s position necessitated diluting the varnish, rotating the
sample, and then reattaching the sample at the exact correct position. A full angular-dependence experiment would
require the sample to be fixed to an appropriate rotator, to avoid the possibility of sample damage.
Differences between the curves with H||TB and H⊥TB of Figs. 2 and 3, as listed above in itens 1 and 3, can be
explained by the following considerations. (1) There is a vortex line along the magnetic field direction; i.e, there are no
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pancake-vortices in the ab planes. (2) Defects located at the boundaries of the twin planes act as pinning centers and
also prevent vortices from crossing the TBs when H||TB. (3) For intermediate and higher fields, the density of vortex
lines pinned with increasing and decreasing applied field is higher for H||TB than for H⊥TB. This final contention can
be obtained after comparison of the values of magnetization in the increasing (decreasing) field of a curve obtained for
H||TB with the correspondent (at same temperature) curve obtained for H⊥TB, as well by comparing the respectives
values of ∆M(H) of both curves. As a consequence of the preceding facts, the average distance between vortex lines,
for intermediate and higher fields, is smaller for H||TB than for H⊥TB.
The vortex-vortex interaction energy [22] is given by F12= (φ2
is the penetration depth, r12is the average distance between vortex lines, and K0is a zeroth-order Hankel function of
imaginary argument. The interaction given by F12is repulsive, and the repulsive force between vortex lines (given by
−∂ F12/∂x for the x direction) increases as r12decreases. Imbalance between −∂ F12/∂x and the magnetic pressure
may produce vortex motion which, in the case of decreasing of the field, produces vortex exit. From the above
considerations, the repulsive force is higher for H||TB than for H⊥TB.
By assuming a triangular lattice of vortex lines (it is also assumed that only the boundary regions of the TBs
plane can stronly pin a vortex) , the repulsion between vortex lines after decreasing the applied magnetic field may
produce vortex motion in two directions approximately perpendicular to each other,and both perpendicular to the
applied magnetic field. When H||TB, one direction of repulsion produces a motion that drives the vortices to cross
the TBs, and the other direction produce a motion that drives the vortices to move between the TBs plane. Then,
when H||TB and the field is decreasing, the TB barriers prevent the vortices crossing the TBs, and the vortex motion
occurs preferentially between the TBs (in this sense, one might consider that the TBs planes act as channels for exit
of vortices). When H⊥TB, the directions of the repulsion between vortex lines are both paralell to the TBs planes,
but in this case, a vortex line is strongly pinned by TBs oriented perpendicular to the vortex line. Furthermore, when
field is decreasing, one may expect that it is easier for vortices to leave the sample for H||TB than for H⊥TB, as
was observed in relaxation data for H = 3 T. Since vortices can exit the sample easily when H||TB, one may expect
that below a certain applied field, the magnetization in the decreasing-field branch for H||TB may eventually become
smaller than the magnetization in the decreasing-field branch for H⊥TB at same temperature, as observed in MvsH
curves of Fig. 2 and Fig. 3.
The values of Hmin, Hon, and Hpen, as obtained from the M(H) curves, are plotted in Fig. 5. The curves drawn
for Hmin and Hon are only to guide the eye. Below 12 K, the values of Hmin increased dramatically for H||TB and
became substantially larger than those for H⊥TB. Above 12 K, the values of Hmin were approximately the same
for both directions of applied magnetic field. The values of Hon were also approximately the same for both field
directions. Below 12 K, the values for H||TB became measurably larger than those for H⊥TB. This trend was the
same as observed for Hmin, but the relative differences were much less.
There was little difference between the values of Hpen for the two field directions. It was found that the values of
Hpen for both field directions could be fitted well by an exponential expression. An exponential response of Hpen
with temperature has been observed by Andrade et al. for anisotropic layered superconductors when H was applied
along the c-axis direction, perpendicular to the ab layers [23]. In that study, the exponential behavior was interpreted
in terms of surface barriers appearing because of the existence of pancake-like vortices lying between the layers. In
the present study, the field was applied along the ab planes and there were no pancake-like vortices lying between the
layers.
The effects of TBs on vortex phenomena can be investigated most directly through dynamics studies. We performed
flux-creep studies, emphasizing the anisotropy of flux dynamics with respect to the TBs. Measurements concentrated
on temperatures below 12 K, because the main effects of the TBs were observed in this temperature regime. The
insets of Fig. 5 show magnetic-relaxation curves, M(t), obtained at 6 K with H = 3 T, for both directions of applied
magnetic field. The curves were obtained for the increasing- and decreasing-field branches and for H = 0 after the field
was discharged (Mrem). All M(t) curves presented an approximately linear response vs. the logarithm of time. It is
interesting to note the large noise in Mrem(t) and also (although not as large) in Mout(t) for H⊥TB. In comparison,
Mrem(t) and Mout(t) were quite consistent for H||TB. Such differences in noise were observed at all temperatures.
The noise in the relaxation measurements seems to be related to the resolution of the measurement, and the fact that
a vortex can exit much more easily when H||TB (large magnetic relaxation) produced less noise in this case. The
insets of Fig. 5 also reveal large differences between Min(t) and Mout(t) for H||TB (but not for H⊥TB). The insets
of Fig. 5 also reveal a large difference in Min(t) and Mout(t) values between H||TB and H⊥TB.
We estimated the current densities, J, at 6 K generated during the initial stage of magnetic relaxation M(t =0)
until M(t = 60-120 s) for the curves shown in the insets of Fig. 5. For a fixed field H applied along the z direction,
in Gaussian units, ∂Bz/∂x = −4πJ/c ∼ 4π∂(M − Meq)/∂x ∼ 4π(dM/dt)(dt/dx) , where Meq is the equilibrium
magnetization, and dx/dt is the flux velocity (which is on the order of cm/s) [22]. Because we did not make local
magnetization measurements (our data were obtained over the entire volume of the sample), we consider that dMeq/dx
∼ 0. From the experimental values of dM/dt (in emu/cm3s) and by assuming dx/dt = 1 cm/s, the estimated values
0/8π2λ2)K0(r12/λ), where φ0is the quantum flux, λ
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of J were:
(1) H||TB: 0.41 A/cm2(flux in), 0.51 A/cm2(flux out), and 0.17 A/cm2(remanent);
(2) H⊥TB: 0.32 A/cm2(flux in), 0.32 A/cm2(flux out), and 0.13 A/cm2(remanent).
A current value of 0.51 A/cm2(the largest current density that was estimated above) corresponds to a transport
current of 5 mA across the largest area of the crystal (1 x 1 mm face), and to 0.5 mA across the smallest area (1 x 0.1
mm face). We note that after 30 min of relaxation, dM/dt decayed to values 30-50 times smaller than those initially
calculated. The flux velocity would also be expected to decay accordingly [24].
Before analyzing the rate of the magnetic relaxation, it is important to obtain the effective activation energy,
U(M). According to Maley et al. [25], U(M) be obtained from U(M)/kB= −Tln|d(M − Meq)/dt| + Tln(Bυa/πd)
, where υ is the attempt frequency, a is the flux hopping distance, and d is the sample thickness. The equilibrium
magnetization Meq is estimated as the average (M++ M−)/2, where M+and M−are respectively the magnetization
in the increasing- and decreasing-field branches of the hysteresis curve [25]. Values of Meq were found to be less
than 10% of M, and we therefore plotted U(M) vs. |M| instead |M − Meq|, for Min(t) and Mout(t) and for both
directions of applied magnetic field (Fig. 6). Each set of data in Fig. 6 reflects a M(t) curve and each point (U(M),
M)) in a set was obtained by first calculating Tln|dM/dt|. The final value of U(M) is obtained by adjusting a value
of the constant C = ln(Bυa/πd) that produced a smooth fit (dotted and full lines in Fig. 6) to the data obtained
for a given configuration and fixed magnetic field. For all curves of Fig. 6, C = 12. This same value of C = 12 was
reported previously for a YBaCuO single crystal [26].
For C = 12, d = 0.1 mm (the thickness of the crystal), and B ∼ 3T, υa ∼ 4.7cm s−1, which is consistent with
a flux hopping distance a = 10 nm and an attempt frequency υ = 4.7x106Hz. Each line in Fig. 6 represents a
fit of U ∼ ln|M|. With the exception of U(Mout) for H||TB, data from 4 to 12 K for the other 3 configurations
fall very close to the respective ln|M| line. Logarithmic decreasing of U with increasing M is consistent with the
linear dependence of M with the logarithm of time, as was observed. Data at T = 2 K, the lowest temperature
of measurement, did not follow the smooth logarithmic fit, and is not shown. Figure 6 reveals that the responses
of U(Min) and U(Mout) were quite similar for H⊥TB, (in the sense that data follow the ln|M| behavior) but not
for H||TB for which only U(Min) follow the logarithimic behavior with M. The differences for H||TB explain the
differences in Min(t) and Mout(t) (and also in the rate of relaxation discussed below) observed for H||TB, as shown
in the inset of Fig. 5.
One may obtain graphically the so-called apparent pinning energy, U0, by constructing a tangent to a given data
set in a given U(M) curve (such as in Fig. 6), where U0is the value at which the tangent intercepts the y axis. This
value can also be obtained by the expression U0= -kBT/S , in which S = (1/M0)(d|M|/dlnt) , and S is the relaxation
rate and M0is the magnetization at time t = 0 [27]. The U0value obtained by such a graphical means is 50% higher
than U0= -kBT/S, which may likely indicate that U0is not well defined for the crystal used in this study.
Because of this consideration, instead of U0, we examined the relaxation rates S for Min (Sin) and Mout (Sout).
The inset of Fig. 6 contains plots of Sin and Sout for both directions of applied magnetic field. In this inset, the
symbols ||in and ||out denote Sin and Sout for H||TB, and ⊥in and ⊥out denote Sin and Sout for H⊥TB. As in Fig.
6 proper, the curves for Sin and Sout for H⊥TB were similar, which provides further evidence of the flux dynamics
being approximately independent of the hysteresis branch (increasing or decreasing) in this test configuration. On the
other hand the responses vs. temperature of Sin and Sout for H||TB confirmed that the barrier for vortices leaving
the sample was lower for this configuration. Comparison between Sout values for both configurations also indicated
that the barrier for vortices leaving the sample was lower for H||TB than for H⊥TB.
The relaxation rates with increasing field in the H||TB configuration were smaller than in the H⊥TB configuration.
This difference might be related to the changes in the position of Hmin observed for H||TB below 12 K.
The differences between various sets of curves disappeared as T approached 12 K; above 12 K, the M(H) curves
were quite similar for both TB configurations. These results suggest that the TBs had a weaker effect on flux dynamics
above 12 K. It is interesting to note that surface barriers are expected to produce a similar asymmetric, but inverted,
responses for Sin and Sout as shown in the inset of Fig. 6 for the configuration H||TB [28].
We finally discuss the flux jumps observed at low temperatures. The data at 2 K (Fig. 3) show the larger magnitude
of the flux jumps when H||TB, and the data at 4 K (inset of Fig. 3), for which a flux jump only occurred in the
hysteresis curve for H||TB, show the same trend. No flux jumps occurred above 4 K (Fig. 2 and other data not
presented in the figures).
It is likely that flux jumps appear due to a continuous imposed dH/dt. On the other hand, data were collected at
a fixed value of H which was reached for a fixed value of dH/dt = 0.02 T/s. In the field region in which flux jumps
were observed, magnetization was measured in intervals of 0.3 T. Upon analyzing the time between two consecutive
data points, with ∆H = 0.3 T, we concluded that the value of 0.02 T/s was not achieved in an interval of 0.3 T.
The built-in program used to charge the magnet probably increased dH/dt to a maximum value, which depended on
∆H, and then rapidly decreased dH/dt as ∆H approached its limit. (In the analysis, we assumed that the ramping
of dH/dt was reproducible for a given ∆H.)
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