Flux creep in Y(Gd) Sa2Cu30~ b. Magnetic field dependence. 1 JUNE 1989. M. Foldeaki, M. E. McHenry,. * and R. C. O'Handley. Department of Materials ...
PHYSICAL REVIEW B
VOLUME 39, NUMBER 16
Flux creep in Y(Gd) Sa2Cu30~
1
JUNE 1989
b. Magnetic field dependence
M. Foldeaki, M. E. McHenry, * and R. C. O'Handley Department
of Materials Science
and Engineering, Massachusetts Institute of Technology, (Received 19 September 1988)
Cambridge,
Massachusetts
02139
The time-dependent was measured after zero-field decay of the diamagnetic magnetization The time dependence was found to cooling (ZFC) over a broad range of fields and temperatures. be linear for short times and logarithmic thereafter within the experimental time intervals for all fields and temperatures. The relaxation rates A(HO, T) and A(H, TO) show maxima at T and respectively (where To and Ho are constant values). The shape of A(H, Tp) is more complicated than that of A(HO, T), which is nearly symmetrical around T . Increasing Ho or To decreases H or T . The activation parameters were found to be strongly field dependent, whereas they agree well for both investigated compositions. The activation energy decreases with increasing field, following the empirical law: Q=a+b/H2. The frequency factor distribution broadens with increasing expression: empirical H, which can be described by the following d g/H. Us— 1n(r02/ro~) ing the activation parameters as determined from A(HO, T) and interpolating by the above expressions, the relaxation rate A(H, TD) can be calculated and agrees well with the experimental data within the experimental error. The data are interpreted using the theoretical model of Beasley, Labusch, and Webb on thermally activated Aux creep over pinning centers assisted by the field-dependent driving force.
H,
I. INTRODUCTION
regarding the physical nature of the process is that it follows first-order kinetics.
Hard superconductors show time-dependent changes in their magnetization (diamagnetic susceptibility) under a magnetic-field gradient. These changes, typically logarithmic in time, ' can be modeled as thermally activated of pinning barriers Auxon motions over a distribution where the apparent activation energy increases as equilibOne of the basic parameters for the rium is reached. practical application of the superconductors is the critical current density. The critical current of hard type-II suthe perconductors by increasing may be increased strength and density of the pinning centers. Measurements of flux flow (through time-dependent magnetic response) probe the flux pinning spectrum and the flux gradient inside the sample. As the Aux gradient depends on the external field, a strong influence of the applied field on the time-dependent response is expected. Activation parameters derived from the time-dependent response characterize the strength of the pinning centers, and, in this way, give information on the flux penetration and exIn the present work only Aux penetration was pulsion. investigated. The activation parameters were determined using the basic theory of thermally activated processes. Other approaches, (e.g. , those based on the theory of hard superconductors) use parameters which can be measured only with a greater experimental error than those of Aux creep. Logarithmic time dependences found in superconductive ceramics are often taken to be indicative of the superconductive glass state or alternatively to be explained When presenting the model. by the Anderson-Kim thermodynamic theory, we show that the. logarithmic time dependence is a consequence of the broad distribution in the activation barrier energies, and the only assumption
II. EXPERIMENTAL METHODS Ceramic samples of Y 1:2:3 were prepared by conventional powder processing from high-purity oxides and carbonates, calcining at 1000 C in air and annealing at 975 C for 5 h followed by a slow cool in Aowing oxygen. X-ray investigation reveals materials of single phase, which are polycrystalline with a grain size of 5 to 50 pm. The samples show a sharp superconducting transition at The 90 K (width 1-3 K) in resistivity measurements. characteristics of these specimens conform in all major respects to those of 1:2:3 ceramics reported in the literature. measurements were carried out on a Magnetization commercial superconducting quantum interference device (SQUID) magnetometer and on a vibrating sample magnetoineter (VSM). Both instruments give essentially the timeequivalent results, but when measuring dependent response, only the VSM allowed the investigation of very short (t (120 sec) time intervals. The magnetization data were taken after zero-field cooling (ZFC) in all cases, i.e. , the applied field was turned on suddenly at the measuring temperatures.
III. EXPERIMENTAL RESULTS A. Field and temperature
'
39
dependence
of the magnetization Figure 1 shows the magnetization MzFc(H, T) measured as a function of the temperature for constant applied field. The field dependence was investigated in detail 11 475
1989
The American Physical Society /
M. FOLDEAKI, M. E. McHENRY, AND R. C. O'HANDLEY
11 476
7.85
39 ~
7.85
'1
1
I
I
1
7.65
E 745—
7.65
7.25—
7P5
-10-
2
1
I
3
4
7.45 -15 0
1
6
7
QPo~ ~o
10
20
30
CD
7. 25
as a function of the temperature, applied fields, Y 1:2:3 sample. Solid lines are guid-
ing the eyes.
only at 1.8, 5, and 10 K (Fig. 2). Data taken on Gd 1:2:3 samples were corrected for the paramagnetic contribution as described in detail elsewhere- and the remaining diamagnetic component shows essentially the same behavior as the Y 1:2:3 samples. The samples were zero-field cooled before each (T, H) point, i.e. , no continuous magnetization curves are given. The behavior shown in Figs. 1 and 2 is typical for ceramic high-T, superconductors.
B. Time-dependent
100
response
MZFc(H, T) is inherently unstable because of the field gradient established at the surface of the superconductor by the sudden application of a magnetic field. If the field is turned on after ZFC, the time dependence of the diamagnetic magnetization can be fairly well described'
400
500
200
Time (seconds
FIG. 1. ZFC magnetization
OO
Y
0
T(K) in diA'erent
1
5
1'n(t}
500
j
FIG. 3. Sample of time dependence of magnetization data & 1 kOe at showing linear and lnt regimes for ZFC Y 1:2:3in T=14.5 K.
0
Deviation from this logarithmic behavior has been observed only for short experimental times (t min) where the time dependence is linear, as predicted theoretically" (Fig. 3). The arbitrary reference time t p is usually 1 sec for the VSM and 1 min for the SQUID. Only the VSM allowed the measurement of the linear portion of the time-dependent response. Focusing first on the dominant logarithmic decay we show in Fig. 4 the logarithmic rate A(HO, T) for Gd 1:2:3 for the constant fields H =1,3, 10 kG for Y 1:2:3. Figure 5 shows A(H, Tp) for the constant temperatures T=1.8, 5, and 10 K. The data of Fig. 4, as well as those published earlier on high-T, sulowextend perconductors well the beyond temperature regime where the relaxation rate is linear' in T. (Doped La2Cu03 does show this linearity at low
(120
) The logarithmic
temperatures.
decay rate A(H, T)
vanishes at T=O, shows a rounded peak, and returns to zero well below the transition temperature. The samples containing Y or Gd do not show any significant differences in fiux creep. The maximum A(T, H) shifts towards
by
m(r) =W(r, )+W (T, H) in(r/«)
10 kG
0.30-
-10
0.20-
0.10-
1.8 K -20
0
10 0.00
0
FIG. 2. ZFC magnetization
as a function of the applied 6eld, at diA'erent constant temperatures. Gd 1:2:3 sample, values corrected for paramagnetic contribution. Solid lines are guiding the eyes.
10
30
20
40
50
T(K)
FIG. 4. Logarithmic rate constant as a function of the temperature, for diA'erent applied fields, Y 1:2:3 sample. Solid lines are guiding the eyes.
FLUX CREEP IN Y(Gd)Ba2Cu307 z.. MAGNETIC FIELD. . .
39
11 477
can be defined as follows: 0.30
P(z)dz
dz/[zln(zq/z~)],
[I/In(z2/z~)]dlnz
if
lnz~
&lnz&lnz2,
"
0.10
0.00— 0
z;
10
zp;exp(Q;/kT) .
The time law for relaxation so derived has the form
H(kg)
FIG. 5. Logarithmic rate constant as a function of the field, for diA'erent temperatures, Gd 1:2:3 sample. Solid lines are guiding the eyes.
M(t, ) —M(t, ) -~M G(t), where
G(t) lower temperatures with increasing fields and its absolute value increases and then saturates.
-I+ [E; ( —t/z
z2)]/ln(z /z ) E; ( —t/—
t/[z) in(z2/z~ )
whereas in the time interval zi & t & z2
A. Relaxation processes following first-order kinetics
6 (t) 1+ (0.5772 —lnz2+lnt
"
Mp
—5M[1 —exp( —t/z)]
(2)
where z is the relaxation time characterizing the process (related to the activation energy Q via the Arrhenius equation: z zpexp(Q/kT), M(t) is the magnetization at the experimental time t, Mp-M(t AM(t —Mp, measures the asymptotic amplitude of the relaxa-
-0),
~)
tion. Not even the simplest classical systems (e.g. , iron with interstitial impurities C or N) follow" Eq. (2) strictly. Usually, instead of well-defined values of z or Q, a distribution has to be supposed. Although assuming a Gaussian distribution would be more physical, usually good results can be obtained by using the so-called "box distribution" of the relaxation times
P(g)-(Q& —g, ) ', Qi&g&Q2, P(Q)
0
Q&Q~
«Q&Q2
(3)
P(g)dg
A transformed
-1.
function
satisfying
.
(9)
While this logarithmic time dependence is quite familiar and can be derived by various methods, the linear time dependence is often overlooked. As shown below, it is important to a complete analysis of the data.
8.
Method of data processing
Comparing Eqs. (1) and (9) we get the following relationship between the experimental rate constant and the relaxation times: A -aM/In(z2/z)
),
(10)
whereas for the linear portion M(t) Mp+at the slope a can be connected to other parameters in the following way:
a
hM/[z~ ln(z2/z~
)]
A/z~,
zi
A/a .
On this basis, we can determine zi by comparing the slopes of the linear and logarithmic portions. To determine z2 from Eq. (10) we need further assumptions for the value of dM because M(~) is not measured. For MzFC(H, tp), 8 0 inside and H H, i,~~;,q outside. The final state for MzFC(H, t) is the Meissner (field cooled) state. As MFC«MzFC for high fields, we can approximate the final state of the zero-field cooled sample by MzFC(H, t ) Consequently, BM-MzFC(H, t p). Using this assumption, Eq. (10) gives
=
ln(z2/z
P(z)
)/ln(z2/zi)
-0.
~
and
,
(7)
],
IV. THEORETICAL MODEL AND NUMERICAL METHOD
The form of the peaks in the relaxation rate in Fig. 4 is very similar to those observed in classical thermally activated reversible or irreversible relaxation processes. In the following section we adopt techniques of relaxation theory to quantify our data. Relaxation processes following first-order kinetics can be described by an exponential time law
),
and E; is the so-called exponential integral. From Eq. (7) Kronmuller" derived approximative expressions for certain time intervals. If t & i],,
G (t)
M(t)
(4)
and P(z)dz 0 otherwise. This is the so-called logarithmic distribution function, introduced by Wiechert for the description of mechanical relaxation and used first by Richter for the analysis of magnetic aftereffect measurements. Further we suppose that the limiting relaxation times z; (i 1, 2) follow the Arrhenius law as well:
0.20
f,, P(z)dz-l
) MzFc(H, tp)/A .
(12)
In this way, we have the values of both of the relaxation
M. FOLDEAKI, M. E. McHENRY, AND R. C. O'HANDLEY times at all temperatures for which A and a are measured. From an Arrhenius plot we can determine the activation parameters Q; and ro;. This evaluation is possible only if the measurements allo~ determination of the short-time region of the curve. Otherwise, an Arrhenius fit based on Eq. (12) [ln(r2/z&) vs 1/T] gives just the width of the activation energy box in the general case. In the special case where Q~ 0, this simplified evaluation results in Q2. In this case it is possible to get an approximate evaluation of the relaxation spectrum based on the value of the logarithmic rate constant A. Both methods of evaluation can be used only at relatively low temperatures (in the 1-kG case K). At higher temperatures, the assumption of up to a logarithmic time law up to infinite times is no longer valid and consequently the evaluated ~2 is incorrect. This is shown empirically by r2 values increasing with temperature. Alternatively, the condition t & r2 for Eq. (9) is not valid at high temperatures. In this case, evaluation can be made only by numerical integration of the function G(r) This requires a considerable amount of computing time without a corresponding increase in the information gained.
-30
C. Data evaluation based on the Anderson-Kim
model
As relaxation proceeds, the system can be modeled as exhausting the low activation energy processes first. Thus, with time, only the larger values of Q remain and the process slows down. Anderson's model of flux creep expressed this situation with a time-dependent activation energy similar to that used in the theory of recovery processes in metals:'"
= —yexp[ —[Qo
dM/dt
P ~M(&) ]l/ka T», I
39
equation. P]otting Mo/A —InA vs 1/T is similar to an Arrhenius plot and results in the activation parameters. V. RESULTS AND DISCUSSION
As a first step, we evaluated the VSM measurements, which include the linear t regime as well as the lnt regime. Figure 6 shows a typical Arrhenius fit for Y 1:2:3,ZFC at 1 kG. Table I summarizes the evaluated activation energies for both samples in 1 kG field. The results demonstrate that Q~-0 in all cases, consequently, SQUID measurements made in fields or temperatures not available in the VSM can be evaluated using Eq. (12) to give the upper limit of the activation energy box. Unfortunately, we get in this case only the width of the box of the preexponential factor, but no information is available on the values of roi or roy because on the basis of the evaluation of the VSM measurements no estimation similarly to Q~-0 can be made on the value of rpi ol' rp2. Figure 7 shows a plot based on Eq. (13) (Anderson-Kim model) for the same sample as that shown in Fig. 6. Results obtained on this basis are also given in Table I for comparison. They clearly show the good agreement between Qo of Eq. (13) and Q2 of Eq. (12). Table II shows the complete set of the data for both samples in all applied fields. The activation energies in Table I do not show any significant dependence on sample composition, whereas the difference in the activation parameters for different applied fields is well beyond the experimental error (+' 2S%). We will now discuss the physical significance of these results.
— —
A. Field dependence of the activation parameters
As shown in Table II, the activation parameters determined from the temperature dependence of the relaxation peaks are strongly field dependent. The activation energy significantly decreases with increasing field. An empirical analysis [Fig. 8(a)] of the data shows that this dependence
I
where
so that as AM(t) I, characterizing the deviation from equilibrium, decreases, the effective activation energy approaches its maximum value, Qo. This maximum value should correspond to the upper limit Qq of our activation energy distribution. The physical significance of y and P will be discussed below. If we neglect transient processes, the integral of this equation results in an equation similar I
to
(1)
M(r)
200
$50upper limit
100-
[lny+Qo/kT+In(kT/P)]kT/P+
(kT/P)lnt . ve rage
(i4) Comparing Eqs. (1) and (14) the following relationship can be found between the Qo y, P, and Mo, A: A kT/P and
Mo
[lny+ Qo/kT+ ln(kT/P)]kT/P .
consequently
inA
ln
y+ Q p/kT,
y is the frequency
1
0.2
0. 1
0.3
1/T(K)
Thus, we can write Mo/A
wer limit
0' 0.0
(is) factor of the activation
FIG. 6. Typical Arrhenius fit for Y 1:2:3,ZFC, 1 kG applied field, resulting in the following activation parameters: Q2 69 meV, Q~ 0.2 meV, ro~ 1.54 sec, r02 0.08 sec. The solid lines correspond to the computer fit.
FLUX CREEP IN Y(od)BaqCu307
39
TABLE I. Activation energies of ZFC relaxation with 1 kG applied Geld evaluated on the basis of the thermodynamical theory (Qi, Q2) or Anderson-Kim model (Qp). For Gd 1:2:3, the magnetization values used in the calculation were corrected for paramagnetic Material
q..
MACxNETIC
FIELD. . .
TABLE II. Activation parameters as a function of the applied field for the Gd 1:2:3 ZFC sample. The magnetization values used in the calculation were corrected for paramagnetic contribution.
contribution. Q2
Gd 1:2:3
—Qi
(meV)
86 69
Y 1:2:3
H (kG)
Qp (meV)
87 67
Q2 (meV)
—14
78 21
1
2 3
6.5
13.5 4. 5
10
can be described by Q2 tt+bH [Fig. 8(a)]. The computer fit on Fig. 8(a) corresponds to a 4, b 74. This means that Q2 has its minimal value for infinite H, and this value is finite, not zero, consequently no fiux penetration can take place without thermal activation even for very high applied fields (values higher than H, 2 should not be supposed). The thermal energy at 10 K is of order 1 meV, giving a good qualitative agreement with the extrapolated 2Q Q2-4 meV. Q diverges at H 0, which is compatible with the fact that there is no fiux penetration without applied field. The frequency factor corresponding to Eq. (12) is strongly field dependent as well, and can be described by the following empirical law: ln(zo2/zoi) ' [Fig. 8(b)]. The computer fit on Fig. 8(b) g —dH corresponds to g 26, d 40. Using these equations to extrapolate the values of the activation energies and frequency factors for di6'erent fields, we calculated the logarithmic rate constant as a function of the applied field at constant temperatures (T 1.8, 4. 2, and 10 K) as follows: in equation ln(z2/zi ) ln(zo2/zoi) + Q2kT (Arrhenius logarithmic form). Substituting the known field dependence for ln(zo2/zoi) and Q2, as well as the measured value of T, we get ln(z2/zi) for all experimental (H, T). Using Eq. (12), we can calculate A from the measured curve using the above calculated semmagnetization
ln(zpi/zpg)
11
22. 5
iempirical values for ln(z2/zi). Figure 9 shows the calculated curves and the experimental points. The agreement is very good at 1.8 K, within 20/o at 10 K, and within 30% even in the worst case, at 4.2 K. The experimental and semiempirical curves have the same shape in all cases. This gives credibility to the empirical evaluation method used.
80
60
40
20
0.0
0.2
o.e
0.4
1.0
1.2
t1/H(k/G)P
200-
1501O
100Cq
Q C
50-
-10
0
0.00
0.05
0. 10
O. f 5
0.20
-20
0.25
FIG. 7. Arrhenius-type plot based on the Anderson-Kim model, Y 1:2:3, ZFC, 1 kG applied field, resulting in Qp 67 meV. For comparison, lnt2lnti vs 1/T, resulting in Q2 —Qi 69 meV, is also sho~n. The solid lines correspond to the computer fit.
0.0
0.2
0.4
0, 6
0.8
1.0
1.2
FIG. 8. Qualitative evaluation of the H dependence of the acparameters: (a) activation energy, (h) preexponential factor. Where applicable, averages of values given in Tables I and II were used. The solid lines correspond to the computer Gt.
tivation
M. FOLDEAKI, M. E. McHENRY, AND R. C. O'HANDLEY
11 480
6 D
0.20
0.10-
0.00
0.0
l
I
I
2.0
4.0
6.0
I
8.0
The observed field dependence cannot be explained by the field dependence of the surface energy barrier ' to fiux the surface as shown in detail elsewhere' penetration barrier disappears between I and 3 kG. Consequently, at higher fields only the pinning barrier, decreased by the driving force, infiuences the activation energy. There are to the field dependence of several theoretical approaches' the driving force (proportional to the fiux gradient), but to use them more detailed information on the nature of the pinning should be necessary. It is clear from the theory, however, that the driving force increases with increasing field. The activation energy we measure arises from thermal activation and driving force simultaneously and a decrease with increasing driving force is compatible with the theories of flux creep. Our experimental result is in good qualitative agreement with the theoretical result of Beasley, Labusch, and Webb, ' who found that the activation energy for fiux creep decreases with increasing fiux gradient for constant 8, and the decrease follows a power law. Although our experiments correspond to the const condition, the results are compatible with the above model because grad8 is a monotonically increasing function of H. The increase of the width of the relaxation time distribution with field as shown in Fig. 8(b) can be qualitatively well understood. With increasing field, the driving force exceeds pinning force for an increasing number of pinning centers, consequently, more pinned fluxoids can escape, giving rise to a broader relaxation time distribution. The difficulties connected to a more quantitative evaluation are increased by the fact that the activation energies obtained from the Arrhenius equation are volume averages, whereas theories usually calculate the activation energy for the unit length of a flux line. As the flux line density depends on temperature, magnetic field, and in a flux creep experiment, on the time, the activation energy per flux line is a more complex quantity than the volume average calculated above, and comparison of experimental and theoretical results can be only approximate. A more detailed theoretical model than used here is beyond the scope of this paper.
—
0.30-
10.0
H(kG) 0.5
0.4-
—
0
0.3-
E 0.2-
—
0.1
0.0
10
0
20
H(kG)
0.30
0.20
0.10
0.00—
0.0
39
2.0
4.0
6.0
8.0
—
10.0
H(kG)
FIG. 9. Comparison of calculated curves (solid lines) of A vs H based on the fits of Fig. 8 and experimental points 8 vs H for (a) T=1.8 K, (b) 4.2 K, and (c) 10 K.
Present address: MST-5, Los Alamos National Laboratory, Los Alamos, NM 87545. 'Y. B. Kim, C. F. Hempstead, and A. R. Strnad, Phys. Rev.
Lett. 9, 306 (1962). 2P. W. Anderson, Phys. Rev. Lett. 9, 309 (1962). 3M. Foldeaki, M. E. McHenry, and R. C. O'Handley, Phys. Rev. 8 39, 2883 (1989). 4C. Ebner and A. Stroud, Phys. Rev. 8 31, 165 (1985). ~K. A. Miiller, M. Takashige, and J. G. Bednorz, Phys. Rev.
ACKNOWLEDGMENTS
This work was supported by a Grant from the Department of Energy DE-FG 02-84ER45174. M. F. was partially supported by the Fulbright Foundation.
Lett. 58, 1143 (1987). M. Foldeaki, M. E.
G. Kalonji, and R. C. Mc Henry, O'Handley, J. Appl. Phys. 64, 5812 (1989). M. E. McHenry, M. Foldeaki, J. McKittrick, R. C. O'Handley, and G. Kalonji, Physica 8 153-155, 310 (1988). M. Foldeaki, M. E. McHenry, J. McKittrick, R. C. O'Handley, and G. Kalonji, Trans. Jpn. Inst. Met. (to be published). A. C. Mota, A. Pollini, P. Visani, K. A. Miiller, and J. G. Bednorz, Phys. Rev. 8 36, 4011 (1987).
39 '
FLUX CREEP IN Y(Gd)Ba2Cu307
M. Tuominen, A. M. Goldman, and M. L. Mecartney, Phys. Rev. B 37, 548 (1988).
''H.
Kronmuller, Nachvvirkttng in Ferromagnetika (SpringerVerlag, Berlin, 1968). A. Nowick and B. Berry, Anelastic Relaxation in Crystalline Solids (Academic, London, 1972). '3P. Gaunt, Philos. Mag. 34, 775 (1976). '4J. W. Martin and R. W. Doherty, Stability of Microstruc tures in Metallic Systems (Cambridge Univ. Press, Cam-
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MAGNETIC FIELD. . .
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