2.2. Magnetization Relaxation
DC magnetization relaxation measurements can provide very useful information on the pinning potential (activation energy), the type of vortex creep (elastic or plastic), and the vortex creep exponent
p that appears in the dependence of the pinning potential on the current density
J [
8]. We have performed a large number of such measurements at various temperatures.
Figure 2 shows such magnetic relaxation measurements, |
m|(
t), at various temperatures and in a DC applied field, μ
0HDC = 1 T, plotted in a double-logarithmic scale, which are clearly straight lines.
For relatively long measurement time
t, the results of DC magnetization relaxation are consistent with thermally activated flux creep [
8]. After the application of a DC external magnetic field
H, and its subsequent removal, considering the parameterization of the pinning potential
U [
9] and the general vortex creep relation [
10], the dependence of the pinning potential on temperature and time-dependent critical current
U[
J(
t),
T] is given by
where
Uc(
T) is a characteristic pinning energy,
Jc0(
T) is the creep-free critical current density, and
t0 is the macroscopic time scale for creep (~10
−6–1 s), which varies weakly with
J(
t). The theoretical models stated that the vortex creep exponent
p is positive in the case of a collective (elastic) vortex creep regime, is negative for plastic creep regime, and depends mainly on the magnetic field
H and the ratio
J/
Jc0. It should be noted that Equation (1) describes both elastic and plastic creep, while in a large number of publications, when dealing with (collective) elastic creep, creep exponent is expressed by μ, notation
p being used only for the plastic creep. A hypothetical fit of the magnetization relaxation curves
mirr(
t) ∝
J(
t) using Equation (1) implies four parameters, clearly too many for acceptable conclusions, and usually results in values of the creep exponent
p that are not consistent with any accepted model. The analysis is much more simplified in the cases in which, for a reasonably long measurement time, the double-logarithmic plots of
mirr(
t) are straight lines, as is evident in
Figure 2. In such cases, one can introduce and calculate a normalized relaxation rate
S = −dln(|
mirr|)/dln(
t) = −dln(
J)/dln(
t) and a normalized pinning potential
U* =
T/
S [
11]. In these conditions, for constant
H and
T, and if the overall
J relaxation is not very high, which is the case of high-performance superconductors,
p and
t0 can be considered constant, and the dependence of the normalized pinning potential on the current density is given by
Following Equations (1) and (2), the normalized pinning potential is
U* =
Uc +
pTln(
t/
t0). For a moderate, window time for measurement and averaging
tw, we can consider that ln(
t/
t0)~ln(
tw/
t0)~constant, and the normalized pinning potential is given by
where
Uc for elastic pinning is lower than the pinning energy for plastic pinning, when the plastic pinning structure can accommodate the vortices. Assuming that ln(
Uc) and ln(
Jc0) depend only weakly on temperature, which is the case for a relatively large
T interval at fixed
HDC, the creep exponent
p can be determined from the results of the magnetization relaxation measurements using Equation (2). For samples with strong pinning, as in our case, the data in the double-logarithmic plot (
Figure 2) are very well described by a straight line with the slope being the normalized relaxation rate
S = −dln(|
m|)/dln(
t). If the time of the measurement
tw is not too large, but is large in comparison with the macroscopic time for flux creep (~10
−6–1 s), to stress the independence of the normalized relaxation rate on time, the normalized relation rate is often noted in the literature as
S* = −Δln(|
m|)/Δln(
t). Using these considerations, we assumed that in the timeframe of our measurements (
t1 = 100 s <
t <
tw = 2700 s), the slope is constant in time. In
Figure 2, it is obvious that the experimental results are straight lines in the double-logarithmic scale. It can also be seen that for a fixed temperature, the time dependence of |
m| ∝
J is very small in comparison with the change due to the variation of temperature, which is important for the determination of creep parameter
p. From the measurements showed in
Figure 2, we have determined for the DC field of 1 T the temperature dependence of the normalized relaxation rate
S = −dln(|
m|)/dln(
t), as shown in
Figure 3, and the normalized pinning potential,
U* =
T/
S, as shown in the inset of
Figure 3. Similar qualitative results were also obtained for other DC magnetic fields and reported elsewhere [
12].
As can be seen in the inset of
Figure 3, the temperature dependence of the normalized pinning potential
U*(
T) has a low-temperature region in which d
U*(
T)/d
T is positive, characteristic of elastic creep, a high-temperature region in which d
U*(
T)/d
T is negative, characteristic of plastic creep, and a maximum that separates the two regions, situated at about 28 K in 1 T. Equation (2), which gives the dependence of the pinning potential on the current density
J ∝ |
m| at a fixed temperature and magnetic field, allows the estimation of the creep exponent
p, if, for not too long measurement time
tw, the decrease in |
m| with time is small compared with the decrease in |
m| with increasing temperature, which is the case for our sample (see
Figure 2). With this approximation, we will consider a
Jav(
T,
H) ∝ |
mav| (
T,
H) averaged over our measurement time. Averaging is made in the double-logarithmic scale, where the dependence is linear, as can be seen in
Figure 2.
If we rewrite Equation (2) as ln(
U*) =
p[ln(1/|
mirr|) +
const], the dependence of ln(
U*) on ln(1/|
mav|) = ln(1/
Jav) taken from the experimental data for each temperature, shown in
Figure 4, allows the distinction between elastic creep and plastic creep, and the determination of the creep exponents.
In
Figure 4, a rounded maximum of ln(
U*) can be seen, which represents a crossover between elastic creep (left-hand side of the graph) and plastic creep (right-hand side). As can be seen from Equation (2), the slopes of the regions in
Figure 4, where ln(
U*) vs. ln(1/|
mav|) dependence is linear, represent the vortex creep exponent
p, which is positive for elastic creep and negative for plastic creep.
2.3. Frequency-Dependent AC Susceptibility
An alternative method of determining the pinning potential in superconductors is by employing the imaginary (out-of-phase) AC susceptibility response of the sample, χ″, as a function of AC field amplitude
hac, for many frequencies of the AC field, at fixed temperatures
T and fixed DC fields μ
0HDC. At a fixed
T, for several μ
0HDC, χ″(
hac) dependence may show a peak at
hac =
h*. The field
h* actually represents the AC field of full penetration of the excitation in the center of the sample, which, in the critical state model, can be correlated with the critical current density of the material [
13]. The positions of the above-mentioned peak depend on the frequency of the AC field excitation due to the different timescale for the vortices to leave the pinning centers. In the case of our CaK1144 single crystal, due to the very high critical current density, the experimental window in which the method is applicable was very close to the critical temperature. In fact, we could obtain useful results only at 35 K very close to the critical temperature, and for four applied DC fields, 0.75, 1, 1.25 and 1.5 T. The results are shown in
Figure 5, the frequencies of the AC excitation fields being indicated in the figure. The selected measurement frequencies were equally distributed, in a logarithmic scale, between a minimum and a maximum one, hence the rather unusual values. It should be noted that the same frequency corresponds to the same symbol and color of the experimental data in any of the four panels of
Figure 5.
The thickness
c of the crystal, 0.04 mm, is much smaller than the other two dimensions, so we could use the Brandt approach on the magnetic flux penetration [
14], which gives the critical current density as a function of the position of the maximum
h* in the χ″(
hac) dependence as
where α ≈ 0.9 is a coefficient that depends slightly on the geometry (rectangle, square or disc), and
c is the thickness, with
h* in Oe,
c in cm, and
Jc in A/cm
2. By using Equation (4), from the
h* values taken from the curves in
Figure 5, we estimated the critical current densities at respective frequencies at 35 K and in the four μ
0HDC mentioned above.
Figure 6 presents the dependence of the critical current density
Jc, shown in logarithmic scale, on ln(
f0/
f), where
f0 is a macroscopic attempt frequency of about 10
6 Hz. The shape of the experimental curves in
Figure 6 indicates which model of pinning is suitable for our sample [
15]: a downward curvature indicates an Anderson–Kim model of pinning with a linear dependence of the pinning potential on the probing current, an upward curvature indicates a collective pinning model with a power-law dependence of the pinning potential on the current, and a straight line in the double-logarithmic plot, as is the case of our measurements, is consistent with a logarithmic dependence of the pinning potential on the probing current density.
As was shown in [
15], for the case of logarithmic dependence of the pinning potential on the current, the slope
b of the straight lines in
Figure 6 is related to the pinning potential
U0 =
kBT(1 + 1/
b). From the slopes in
Figure 6, we estimated the resulted values of
U0 in K (
kB = 1), for each μ
0HDC: 260 ± 5 K (0.8 T), 210 ± 5 K (1 T),170 ± 5 K (1.25 T), and, respectively, 150 ± 5 K (1.5 T).
2.4. Magnetic Hysteresis Loops and Critical Current Density
Considering the high values of pinning potential in a wide temperature range, with values higher than 1000 K around the temperature of liquid hydrogen, it is also important to evaluate the critical current density
Jc. For this purpose, we have performed magnetization hysteresis measurements to determine the field dependence of the irreversible magnetization, at various temperatures. Such measurements are shown in
Figure 7, in which, for clarity, we present the results in three panels, with different scales for the magnetic moment
m(
HDC,
T), for the increasing and, respectively, decreasing applied magnetic field.
Several things can be seen in
Figure 7. First, in increasing field branches at low fields, the Meissner screening is evident, followed by penetration of vortices, with diamagnetic response decreasing in absolute values. Then, at high temperatures (top panel) for fields higher than a certain value (irreversibility field,
Hirr), magnetization is zero for both increasing (
H)↑ and decreasing (
H)↓ applied field. Also, a small Second Magnetization Peak (SMP) can be seen, which is a nonmonotonic variation of Δ
m(
H) =
m(
H)↓ −
m(
H)↑. At an intermediate temperature, SMP is also very well seen, with a quite flat dependence of Δ
m(
H) on the temperature. At low temperatures (bottom panel), SMP disappears, and Δ
m(
H) has the normal decrease with increasing temperature and field.
Neglecting the low field data (for fields smaller than the lower critical field
Hc1), from the Δ
m(
H) data we can estimate the critical current density as a function of temperature and field, using the modified Bean critical state model [
13]. With the dimensions of the rectangle (the face of the sample perpendicular to the magnetic field)
l (length) and
w (width), with
l >
w, and the sample dimension in the field direction
d (thickness), the critical current density, in A/cm
2, is given by
with Δ
m(
H) =
m(
H)↓ −
m(
H)↑ in emu, and all three dimensions in cm. If the sample is a square plate, i.e.,
l =
w (which is the case of our sample), Equation (5) becomes
Using the data in
Figure 7 and Equation (6), we evaluated the values of the critical current density as a function of field and temperature
Jc (
H,
T), as shown in the 3-dimensional plot in
Figure 8.
In
Figure 8 we can clearly see the surface showing the second magnetization peak, as well as the quite high values of the critical current density, for temperatures not too close to the critical temperature, and for a wide range of applied magnetic fields. For temperatures around the temperature of liquid hydrogen (20 K),
Jc is of the order of 10
5 A/cm
2 even at the highest field of our measurements, which is quite important for future applications.