- Email: [email protected]
Pinning forces in Nb and InPb type II superconductors
Physica I07B (1981) 467-468 North-Holland Publishing Company
IA 7
PINNING FORCES IN Nb AND In-Pb TYPE II SUPERCONDUCTORS C S CHOU
, J F ALLEN,
and J G M ARMITAGE
School of Physical Sciences, Scotland, U K *Department
University of St. Andrews
of Physics, Chinese Culture University Taipei, Taiwan, China
The pinning forces in Nb and Pb-In were determined by mechanically sweeping a magnetic field over it and measuring the resulting force. Three different forces due to the interaction between the magnetic field ,the trapped flux lines, the Meissner screening current, and the pinning sites of the specimen were identified. The magnitude of the bulk pinning force was in agreement with that calculated from the magnetization via Irie-Yamafuji model; F = ~ B ~ . The temperature dependence of the pinning force was found to be given by Fp(T)=Fp(0)(I-T/T¢) and it is suggested that this can be understood in terms of Anderson's flux creep model. and I. INTRODUCTION Fp= 64.1 B 0-79 dyne at 1.71K. The pinning forces were measured by the The value of Fp for the mechanically direct measurement of the force exerted polished NbII is on the superconducting specimen when a non-uniform magnetic field was swept FD=43.5 B 0"87 dyne at 4.22K, past it. The resulting forces on the for %he oxidized Nbll is specimen depend in a complex way on the Fp=43.6 B 0.80 dyne at 4.22K, magnitude and non-uniformity of the and for the chemically polished Nbll is sweeping field in the neighbourhood of the specimen and on the magnetic state Fp=43.6 B 0"73 dyne at 4.22K. of the specimen. We interpret those The value of V in the above expressions forces in terms of that due to the inwere determined from the remanent field teraction between the magnetic field, • The value of ~ probably depends only the trapped flux lines, the Meissner on the density and distribution of the screening current~ and the pinning pinning sites. The values of the coesites of the specimen; FA, F s and Fp. fficient ~ , which varies with temperature, determined by 2. EXPERIMENTAL RESULTS The F n forces measured on two 3ram long o< = F pmax/BHRma x . x 6.35mm dia. niobium specimens NbI and ~;e have used the experimental maximum NbII, at different temperatures and pinning force FDmax and the magnetic with different surface treatments are induction B H R m ~ of the specimen at plotted in Figure i. The results of F A HRmax, where BH. is the bulk maxiand F s is discussed elsewhere I. mum remanent f i e ~ X a n d HRmax is the 3. DISCUSSIONS corresponding maximum field of the nonuniform eaternal field profile. 3.1 The bulk pinning force On the Irie-Yemafuji 2 model, the bulk pinning force is given by Fp=~B'. The results of our measurements of F o with the chemically polished NbI and ~he mechanically polished, oxidized, and chemically polished NbII specimens are compared with the Irie-Yamafuji model in Figure i. The d o t t e d lines are calculated from the magnetization curve measurements and the dashed lines are from our experimental results. The value of Fp for the chemically polished NbI specimen at various temperatures are Fp= 55.5 B 0-79 dyne at 4.22K, Fp= 58.9 B 0"79 dyne at 3.75K, Fp= 61.7 B0-79dyn e
at 2.95K,
Fp= 63.9 B 0-79 dyne at 2.06K,
03784363/81/0000-0000/$02.50
© North-HollandPublishingCompany
3.2 Temperature dependence pinning force Fp
of the bulk
The maximum pinning force Fpm~x in different specimens plotted as function of the reduced temperature T/Tc, are shown in Figure 2. The results can be well-represented by a remarkably simple equation Fpmax(T)= Fpmax(0) (I-T/Tc) , 1 where Fpmax(0) is the maximum bulk pinning force of the specimen at zero temperature. The magnitudes of Fnmax(0) for different specimens were =~erived by extrapolating the curves in Figure 2 to intersect the y-axis, and are specimen Fpmax(0) dyne Tc K Mech. pol. NbI 38.4 x 103 9.28 Chem. pol. NbI 30.9 x 103 9.28 Mech. pol. NbII 65.2 x 103 9.28 467
468
Oxidized NbII Chem. pol.NbII Pb93In7 Pb90Inl0
42.4 20.8 7.3 6.1
x x x x
103 103 10 3 103
9.28 9.28 6.5 6.4 I
Hamax x 229 Oe
3.~3~
U~........ j - f .--
,
I
mech. pol. NbI
c6 ~0
% I
:
I lmF
a 4.221<
~s4
b
~ " 8
~'I~ / /~- ~ "
~"<"~ -.-:::::-:'..~"
3.75K
C 2.95K d 2.44K e 2.09K
6 .....". . ~ . . - ~ .....:'..."
12
Chem. pol. Nbl
l
Chem] pol.
J
....w
oxidized
m
. pol.
4.22
,
F x 1410 dyne
0
FD= g - kT In(R0/R c) (lvD) -1 . 3 If w~e assume that R c and 1 ake independent of temperature, we have Fp(0)= g ( i v D ) - I 4 at T=0. Since Pp(Tc)=0 and (ivp)-l#0 at T=Tc, we have g=kT c in(R0/Rc). 5 Rewriting Equation 3 as
~
.-
3'
If we take Anderson's3 original concept of a pinning barrier which was that the flux lines were trapped in a potential well of depth g, the rate at which they can escape from the pinning barrier can be expressed in terms of an activation energy U, R=R0 exp(-U/kT), 2 where R 0 is the natural frequency of vibration of the flux line and k is the Boltzmann constant. The activation energy U of a flux bundle may be written U = g - VpFl , where vp is-the bundle volume, F is the gradien% of the magnetic pressure which is equal to Fp of the pinning site at the critical point just before the release of the bundle, and 1 is some characteristic length relating the force to an energy. If the rate of escape of the flux from the potential well at F=FDmax is designated as Rc, Fietz4and Webb~have given
9m
6'
I
1'2
15
1'8
21,
J 27
2'4
Fp(T)=
g - kT c in(R0/Rc)~c
and substituting ,we obtain
(iVp) -I, 6
Equation 4 &5 into Eq.6
F~(T)= FD(0)(I-T/Tc). Thi~ resul~ is in agreement with our experimental observations.
Figure 1 Fp vs Hamax of NbI & NbII REFERENCES:
Fpmax~410
24
dyne
~- ~
a b c d e f
\
pol. pol. pol. NbII pol.
NbI NbI NbII NbII
Pb93In 7
g Pb90Inl0
--.
\
-
- -----L2
0
mech. chem. chem. oxid. mech.
0"2
"
-
"-.,.....
~
0.4
J
,,
L
'
" -
0.6
Figure 2 Temperature Fpmax of different
¢-
0.8
dependence specimens
\
1.0 of
l.Chou C S, 1979, Ph.D. Thesis (Univ. of St. Andrews, Unpublished). 2. Irie F, and Yamafuji K, 1967, J. Phys Soc. Japan, 23, 255. 3. Anderson P W, 1962, Phys. Rev. Lett., 9, 309. 4. Fietz W A, and Webb W W, 1969, Phys. Rev., 178, 657.
IA 7
PINNING FORCES IN Nb AND In-Pb TYPE II SUPERCONDUCTORS C S CHOU
, J F ALLEN,
and J G M ARMITAGE
School of Physical Sciences, Scotland, U K *Department
University of St. Andrews
of Physics, Chinese Culture University Taipei, Taiwan, China
The pinning forces in Nb and Pb-In were determined by mechanically sweeping a magnetic field over it and measuring the resulting force. Three different forces due to the interaction between the magnetic field ,the trapped flux lines, the Meissner screening current, and the pinning sites of the specimen were identified. The magnitude of the bulk pinning force was in agreement with that calculated from the magnetization via Irie-Yamafuji model; F = ~ B ~ . The temperature dependence of the pinning force was found to be given by Fp(T)=Fp(0)(I-T/T¢) and it is suggested that this can be understood in terms of Anderson's flux creep model. and I. INTRODUCTION Fp= 64.1 B 0-79 dyne at 1.71K. The pinning forces were measured by the The value of Fp for the mechanically direct measurement of the force exerted polished NbII is on the superconducting specimen when a non-uniform magnetic field was swept FD=43.5 B 0"87 dyne at 4.22K, past it. The resulting forces on the for %he oxidized Nbll is specimen depend in a complex way on the Fp=43.6 B 0.80 dyne at 4.22K, magnitude and non-uniformity of the and for the chemically polished Nbll is sweeping field in the neighbourhood of the specimen and on the magnetic state Fp=43.6 B 0"73 dyne at 4.22K. of the specimen. We interpret those The value of V in the above expressions forces in terms of that due to the inwere determined from the remanent field teraction between the magnetic field, • The value of ~ probably depends only the trapped flux lines, the Meissner on the density and distribution of the screening current~ and the pinning pinning sites. The values of the coesites of the specimen; FA, F s and Fp. fficient ~ , which varies with temperature, determined by 2. EXPERIMENTAL RESULTS The F n forces measured on two 3ram long o< = F pmax/BHRma x . x 6.35mm dia. niobium specimens NbI and ~;e have used the experimental maximum NbII, at different temperatures and pinning force FDmax and the magnetic with different surface treatments are induction B H R m ~ of the specimen at plotted in Figure i. The results of F A HRmax, where BH. is the bulk maxiand F s is discussed elsewhere I. mum remanent f i e ~ X a n d HRmax is the 3. DISCUSSIONS corresponding maximum field of the nonuniform eaternal field profile. 3.1 The bulk pinning force On the Irie-Yemafuji 2 model, the bulk pinning force is given by Fp=~B'. The results of our measurements of F o with the chemically polished NbI and ~he mechanically polished, oxidized, and chemically polished NbII specimens are compared with the Irie-Yamafuji model in Figure i. The d o t t e d lines are calculated from the magnetization curve measurements and the dashed lines are from our experimental results. The value of Fp for the chemically polished NbI specimen at various temperatures are Fp= 55.5 B 0-79 dyne at 4.22K, Fp= 58.9 B 0"79 dyne at 3.75K, Fp= 61.7 B0-79dyn e
at 2.95K,
Fp= 63.9 B 0-79 dyne at 2.06K,
03784363/81/0000-0000/$02.50
© North-HollandPublishingCompany
3.2 Temperature dependence pinning force Fp
of the bulk
The maximum pinning force Fpm~x in different specimens plotted as function of the reduced temperature T/Tc, are shown in Figure 2. The results can be well-represented by a remarkably simple equation Fpmax(T)= Fpmax(0) (I-T/Tc) , 1 where Fpmax(0) is the maximum bulk pinning force of the specimen at zero temperature. The magnitudes of Fnmax(0) for different specimens were =~erived by extrapolating the curves in Figure 2 to intersect the y-axis, and are specimen Fpmax(0) dyne Tc K Mech. pol. NbI 38.4 x 103 9.28 Chem. pol. NbI 30.9 x 103 9.28 Mech. pol. NbII 65.2 x 103 9.28 467
468
Oxidized NbII Chem. pol.NbII Pb93In7 Pb90Inl0
42.4 20.8 7.3 6.1
x x x x
103 103 10 3 103
9.28 9.28 6.5 6.4 I
Hamax x 229 Oe
3.~3~
U~........ j - f .--
,
I
mech. pol. NbI
c6 ~0
% I
:
I lmF
a 4.221<
~s4
b
~ " 8
~'I~ / /~- ~ "
~"<"~ -.-:::::-:'..~"
3.75K
C 2.95K d 2.44K e 2.09K
6 .....". . ~ . . - ~ .....:'..."
12
Chem. pol. Nbl
l
Chem] pol.
J
....w
oxidized
m
. pol.
4.22
,
F x 1410 dyne
0
FD= g - kT In(R0/R c) (lvD) -1 . 3 If w~e assume that R c and 1 ake independent of temperature, we have Fp(0)= g ( i v D ) - I 4 at T=0. Since Pp(Tc)=0 and (ivp)-l#0 at T=Tc, we have g=kT c in(R0/Rc). 5 Rewriting Equation 3 as
~
.-
3'
If we take Anderson's3 original concept of a pinning barrier which was that the flux lines were trapped in a potential well of depth g, the rate at which they can escape from the pinning barrier can be expressed in terms of an activation energy U, R=R0 exp(-U/kT), 2 where R 0 is the natural frequency of vibration of the flux line and k is the Boltzmann constant. The activation energy U of a flux bundle may be written U = g - VpFl , where vp is-the bundle volume, F is the gradien% of the magnetic pressure which is equal to Fp of the pinning site at the critical point just before the release of the bundle, and 1 is some characteristic length relating the force to an energy. If the rate of escape of the flux from the potential well at F=FDmax is designated as Rc, Fietz4and Webb~have given
9m
6'
I
1'2
15
1'8
21,
J 27
2'4
Fp(T)=
g - kT c in(R0/Rc)~c
and substituting ,we obtain
(iVp) -I, 6
Equation 4 &5 into Eq.6
F~(T)= FD(0)(I-T/Tc). Thi~ resul~ is in agreement with our experimental observations.
Figure 1 Fp vs Hamax of NbI & NbII REFERENCES:
Fpmax~410
24
dyne
~- ~
a b c d e f
\
pol. pol. pol. NbII pol.
NbI NbI NbII NbII
Pb93In 7
g Pb90Inl0
--.
\
-
- -----L2
0
mech. chem. chem. oxid. mech.
0"2
"
-
"-.,.....
~
0.4
J
,,
L
'
" -
0.6
Figure 2 Temperature Fpmax of different
¢-
0.8
dependence specimens
\
1.0 of
l.Chou C S, 1979, Ph.D. Thesis (Univ. of St. Andrews, Unpublished). 2. Irie F, and Yamafuji K, 1967, J. Phys Soc. Japan, 23, 255. 3. Anderson P W, 1962, Phys. Rev. Lett., 9, 309. 4. Fietz W A, and Webb W W, 1969, Phys. Rev., 178, 657.