- Email: [email protected]
Anomaly in magnetic behavior in oxide superconductors
PMm
Physica C 185-189 (1991) 2333-2334 North-Holland
ANOMALY IN MAGNETIC BEHAVIOR IN OXIDE SUPERCONDUCTORS Teruo MATSUSHITA, Edmund Soji OTABE and Baorong NI* Department of Computer Science and Electronics, Kyushu Institute of Technology, Iizuka 820, Japan. *Department of Electronics, Kyushu University, Fukuoka 812, Japan It has been observed in measurements of imaginary ac susceptibility, X", of oxide superconductors vs temperature that a peak is shifted to higher temperature with a reduction in its value according to decreasing ac field amplitude. The anomalous reduction in the peak value contradicts the theoretical prediction by the critical state model and is caused by a reversible motion of fluxoids in pinning potential. In this paper X# is calculated on the basis of the Campbell model. 1. INTRODUCTION It has been known that magnetic behaviors of oxide superconductors can be described well by the critical state model in which the pinning force on fluxoids is assumed to be completely irreversible. This model predicts that the imaginary ac susceptibility, X", has a peak at the ac field amplitude, Hm, approximately equal to the penetration field, Hp = J,d[2, where J¢ is the critical current density and d is the sample size) Such peaks are usually observed in X" vs temperature curve and these results suggest the validity of the model. However, values of the peak depends on H~ and sometimes take much smaller value than the prediction, (3/4~r)~0. That is, the loss energy is much smaller than predicted. This suggests that the phenomenon is affected by reversible fluxoid motion in pinning potentials 2 which is pronounced in samples smaller than the ac penetration depth, A0.3 Since electromagnetic properties in the regime of reversible fluxold motion can be described by the Campbell model, 2 the imaginary ac susceptibility is calculated by using the this model in this paper. 2. THEORY It is assumed that the dc and ac magnetic fields, Ha + L ~ sinwQ are applied along the z-axis to an intinite superconducting slab which exists in the region of 0 _< x _< d. The critical current density can be regarded as constant at a given dc field and temperature. It is also assumed that the flux distribution is initially (wt = -~r/2) in the critical state. The variatlon in the surface field causes the displacement of the fluxoids,
u(x), in the superconductor and the resultant variation in the magnetic flux density, b(x). These quantities are related through the conth:uity equation: =
(l)
l~oHo
dz
The variation in the flux density leads to a variation in the Lorentz force density, -Hodb(x)/dx. This is balanced with the pinning force density, given by 2 - - 2:,oJ.
to
[
! - exp
,
(Z)
where d, is the interaction distance representing a half size of the pinning potential. From Eqs.(1) and(2) we have
with Ao =
(HodJ3~) ~n.
This equation can be solved nu-
merically with the boundary conditions of b(0) = ttoH,~(1
+ sinw~) and db/dz = 0 at x = d/2. Then, the imaginaxy susceptibility, X#, is calculated from , 1 Jo ,b} cos ~ d ~ , X = ~H,,
(4~
where ( 1 represents an average in the superconductor. An example cf numerically calculated re.suit for d much smaller than A0 is shown in Fig. 1. ]t is seen that deviatioi~ from the critical st.ate mede~ at low H~'s is remarkable and the peak of X;" ~s shifted ',o higher 2:.:~ wi~ a reduction of the peak vMue. Since the above numerica! calcuRation is complicated. an approximate simple expression of ~" is proi~os~ here: X"
092D4534t~b'$03.50 © i99i - Elsevier Science Publishers B.V. Aft fights resewed.
2~0
::pH~
311 + z(2A0/d}~l°Sr: 4- ::2
'.~'i
T. Matsushitaet aL/ AnomMy in magneticbehaviorin oxidesuperconductors
2334
....... I
........
I
. . . . . . . .
........
I
........
I
I
......
/°'~
0
10 -1
/,
..--... 10 10 -a
/.t"
/
%. N,
/ o/*
%,\
l
1
/~ 0Hm=5.0 raT ..'" •"
]
"-..
3.0
'".
o 0.1
~L
.-'C x.
1.0
.'" ."
/'/'/°
.... ""
l
k
.'" -'" ...................
t ......
10 -4
.......
I
........
I
,
10 -2
1
0.5
10 a
Hm/Hp FIGURE
t
FIGURE 2
I
X"/Ito vs H,,,/H~, for the case of d/2Au = 0.3 obtained from numerical calculation ( - - ) , state model ( ~ - ~ - ) and Eq. (5) (-. . . . . . . . . -).
1
X"/Po vs t for powdered Y-Ba-Cu-O at various H~'s.
critical
I
This approaches X" = (l~oH,,,16rHp)(d/2Ao)' in the reversible limit of d << A0 and small H,,,'s. In the opposite case of d >> Ao, Eq. (5) agrees also with the
:::l
I
I
~
I
I
I
I
I
0.1
prediction by the critical state model both in small and large limits of H , , This expression is compared with the result of numerical calculation in Fig. 1. It is seen that Eq. (5) approximates well the numerically calculated resuit. Here we assume the tempexature dependence of J~ and Ao as d~(t) = de(0)(1 - t2) TM,
0.5
1
A0(t) = A0(0)(1 - t2) -", (6)
where t = T/To is the reduced temperature and m and n are parameters. Then, the X" vs temperature curve can be easily obtained from Eq. (5).
FIGURE 3 Calculated X"/po vs t from Eq. (5).
or larger than the specimen size. This can be seen from
dered Y-Ba-Cu-O with a mean size of 8.0 ~tm. Calculated X" vs t curves for various Hm's are given in Fig. 3, where we have assumed Jo(0) = 3 x 109 A/m 2, A0(0) = 0.6 gin,
a peak value of X". That is, if the peak value of X" is much smaller than (3/4zr)#o, the reversible phenomenon is dominant. It should be noted that in this case estimation of Jc from X"-peak in terms of the critical state model is misleading
m = 5.0 and n = 3.0 and substituted the mean powder size in d. These results show that the peak shifts te
REFERENCES
3. DISCUSSION Figure 2 represents experimental results of X" on pow-
higher t accompanied by a reduction in the peak value according to a decrease in Hm. Qualitative agreement between calculation and experiments is satisfactory. In conclusion the reversible phenomenon influences much the imaginary ac susceptibility when Ao is comparable to
1. T. Matsushita and B. Ni, Jpn. J. Appl. Phys. 28 (1989) L419. 2. A. M. Campbell, d. Phys. C 4 (197:[) 3186. 3. F. Sumiyoshi, M. Matsuyama et af., Jpn. Phys. 25 (1986) Lt48.
J. Appl.
Physica C 185-189 (1991) 2333-2334 North-Holland
ANOMALY IN MAGNETIC BEHAVIOR IN OXIDE SUPERCONDUCTORS Teruo MATSUSHITA, Edmund Soji OTABE and Baorong NI* Department of Computer Science and Electronics, Kyushu Institute of Technology, Iizuka 820, Japan. *Department of Electronics, Kyushu University, Fukuoka 812, Japan It has been observed in measurements of imaginary ac susceptibility, X", of oxide superconductors vs temperature that a peak is shifted to higher temperature with a reduction in its value according to decreasing ac field amplitude. The anomalous reduction in the peak value contradicts the theoretical prediction by the critical state model and is caused by a reversible motion of fluxoids in pinning potential. In this paper X# is calculated on the basis of the Campbell model. 1. INTRODUCTION It has been known that magnetic behaviors of oxide superconductors can be described well by the critical state model in which the pinning force on fluxoids is assumed to be completely irreversible. This model predicts that the imaginary ac susceptibility, X", has a peak at the ac field amplitude, Hm, approximately equal to the penetration field, Hp = J,d[2, where J¢ is the critical current density and d is the sample size) Such peaks are usually observed in X" vs temperature curve and these results suggest the validity of the model. However, values of the peak depends on H~ and sometimes take much smaller value than the prediction, (3/4~r)~0. That is, the loss energy is much smaller than predicted. This suggests that the phenomenon is affected by reversible fluxoid motion in pinning potentials 2 which is pronounced in samples smaller than the ac penetration depth, A0.3 Since electromagnetic properties in the regime of reversible fluxold motion can be described by the Campbell model, 2 the imaginary ac susceptibility is calculated by using the this model in this paper. 2. THEORY It is assumed that the dc and ac magnetic fields, Ha + L ~ sinwQ are applied along the z-axis to an intinite superconducting slab which exists in the region of 0 _< x _< d. The critical current density can be regarded as constant at a given dc field and temperature. It is also assumed that the flux distribution is initially (wt = -~r/2) in the critical state. The variatlon in the surface field causes the displacement of the fluxoids,
u(x), in the superconductor and the resultant variation in the magnetic flux density, b(x). These quantities are related through the conth:uity equation: =
(l)
l~oHo
dz
The variation in the flux density leads to a variation in the Lorentz force density, -Hodb(x)/dx. This is balanced with the pinning force density, given by 2 - - 2:,oJ.
to
[
! - exp
,
(Z)
where d, is the interaction distance representing a half size of the pinning potential. From Eqs.(1) and(2) we have
with Ao =
(HodJ3~) ~n.
This equation can be solved nu-
merically with the boundary conditions of b(0) = ttoH,~(1
+ sinw~) and db/dz = 0 at x = d/2. Then, the imaginaxy susceptibility, X#, is calculated from , 1 Jo ,b} cos ~ d ~ , X = ~H,,
(4~
where ( 1 represents an average in the superconductor. An example cf numerically calculated re.suit for d much smaller than A0 is shown in Fig. 1. ]t is seen that deviatioi~ from the critical st.ate mede~ at low H~'s is remarkable and the peak of X;" ~s shifted ',o higher 2:.:~ wi~ a reduction of the peak vMue. Since the above numerica! calcuRation is complicated. an approximate simple expression of ~" is proi~os~ here: X"
092D4534t~b'$03.50 © i99i - Elsevier Science Publishers B.V. Aft fights resewed.
2~0
::pH~
311 + z(2A0/d}~l°Sr: 4- ::2
'.~'i
T. Matsushitaet aL/ AnomMy in magneticbehaviorin oxidesuperconductors
2334
....... I
........
I
. . . . . . . .
........
I
........
I
I
......
/°'~
0
10 -1
/,
..--... 10 10 -a
/.t"
/
%. N,
/ o/*
%,\
l
1
/~ 0Hm=5.0 raT ..'" •"
]
"-..
3.0
'".
o 0.1
~L
.-'C x.
1.0
.'" ."
/'/'/°
.... ""
l
k
.'" -'" ...................
t ......
10 -4
.......
I
........
I
,
10 -2
1
0.5
10 a
Hm/Hp FIGURE
t
FIGURE 2
I
X"/Ito vs H,,,/H~, for the case of d/2Au = 0.3 obtained from numerical calculation ( - - ) , state model ( ~ - ~ - ) and Eq. (5) (-. . . . . . . . . -).
1
X"/Po vs t for powdered Y-Ba-Cu-O at various H~'s.
critical
I
This approaches X" = (l~oH,,,16rHp)(d/2Ao)' in the reversible limit of d << A0 and small H,,,'s. In the opposite case of d >> Ao, Eq. (5) agrees also with the
:::l
I
I
~
I
I
I
I
I
0.1
prediction by the critical state model both in small and large limits of H , , This expression is compared with the result of numerical calculation in Fig. 1. It is seen that Eq. (5) approximates well the numerically calculated resuit. Here we assume the tempexature dependence of J~ and Ao as d~(t) = de(0)(1 - t2) TM,
0.5
1
A0(t) = A0(0)(1 - t2) -", (6)
where t = T/To is the reduced temperature and m and n are parameters. Then, the X" vs temperature curve can be easily obtained from Eq. (5).
FIGURE 3 Calculated X"/po vs t from Eq. (5).
or larger than the specimen size. This can be seen from
dered Y-Ba-Cu-O with a mean size of 8.0 ~tm. Calculated X" vs t curves for various Hm's are given in Fig. 3, where we have assumed Jo(0) = 3 x 109 A/m 2, A0(0) = 0.6 gin,
a peak value of X". That is, if the peak value of X" is much smaller than (3/4zr)#o, the reversible phenomenon is dominant. It should be noted that in this case estimation of Jc from X"-peak in terms of the critical state model is misleading
m = 5.0 and n = 3.0 and substituted the mean powder size in d. These results show that the peak shifts te
REFERENCES
3. DISCUSSION Figure 2 represents experimental results of X" on pow-
higher t accompanied by a reduction in the peak value according to a decrease in Hm. Qualitative agreement between calculation and experiments is satisfactory. In conclusion the reversible phenomenon influences much the imaginary ac susceptibility when Ao is comparable to
1. T. Matsushita and B. Ni, Jpn. J. Appl. Phys. 28 (1989) L419. 2. A. M. Campbell, d. Phys. C 4 (197:[) 3186. 3. F. Sumiyoshi, M. Matsuyama et af., Jpn. Phys. 25 (1986) Lt48.
J. Appl.