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The maximum possible loss-free current of type-II superconductors with flux pinning
Volume 77A, number 6
PHYSICS LETTERS
23 June 1980
THE MAXIMUM POSSIBLE LOSS-FREE CURRENT OF TYPE-lI SUPERCONDUCTORS WITH FLUX PINNING E.H. BRANDT Max-Planck-Institut fiir Meraliforsehung, Inst itut ftir Physik, Stuttgart, Germany Received 27 March 1980
The “dilute limit” and the “pinning threshold” are shown not to exist. An upper limit for the critical current in irreversible type-LI superconductors is given.
The maximum loss-free current density, ~ a typeII superconductor can support due to pinning of flux lines at material inhomogeneities is as important a material property as the transition temperature and the upper critical field H~2(O).Although there exists a large number of experimental and theoretical papers on/a [1] we are far from an understanding of the observed critical currents even in cases where the metallurgical structure of the superconductor is known. The problem is not so much the elementary pinning forces, which in many cases can be estimated satisfactorily from the Ginsburg—Landau and BCS theories, It is rather the summation of these forces to a volume pinning force F~= JcB. There are essentially three different methods, forp the specialsummation case of identical pointwhich pins we withdiscuss density and with maximum pinning force f~. 1. If the flux-line lattice (FLL) is soft it can adjust to the pins and each pin exerts its maximum force.
uli of the FLL. As discussed in detail by Kramer [5] and Campbell [6] the dilute limit formula does not explain the experiments. Even if the threshold is ignored (this was done in many papers) and even if the formula is improved to account for the periodicity of pinning forces due to the periodic FLL, the theoretical values for F~are still too small by several orders of magnitude. In fact the experimental values are closer to the direct summation (1). There are various reasons for the failure of formula (1). The most important one is that the “dilute limit” does not exist. Dilute means that, at the position of one pin, the flux-line displacements s caused by other pins may be neglected. This condition is never satis2) infied for linearly large specimens the mean square creases with the since specimen diameter D. (s This fact is due to the slow spatial decrease of the displacement field caused by one pin. One has [81 (f2> IC
2 +y2) + z2)\ 1/2 (3)
44
This is the case of direct summation, F —pf. P
(Is(r) —s(O)I 2> = (1)
p
2. The summation formula applied most often was derived in the “dilute limit” (p 0 but still a large number of pins) [2—41: F = 0 f ~fti~ r (2) -÷
p
thr’
p
>
thr
where fthr d2(c,~c 1/2is a threshold force, d is 66) c the flux-line spacing, and 44 and c66 are elastic mod484
(x
C “ 44C66
C66
2
2
where r = (x;y; z) and (f ) for periodic pinfling forces. Thus, if the FLL is shifted across a sufficiently large array of dilute pins the FL displacements at a given pin~‘deven oscillate for statistically with ampli2>1!2 weak pins. Thislarge means there tude is no (sthreshold. The divergence of (s ) in the limit D -÷°° was removed in ref. [2] by introduction of the Labusch parameter ~L’ which leads to an exponential decrease of s at larger distances. However, one can show that with-
Volume 77A, number 6
PHYSICS LETTERS
23 June 1980
in Labusch’s theory, below the threshold and for arbitrary shape of the pinning potential Unot only does FP=(—VU>vanishbutalsotheparameteraL=(V2U) (iF the averaging is performed properly [7]). With the consistent value = 0 inserted, various expressions in ref. [2] diverge and loose their sense. A correct theory should give aL ~ 0 and F~ 0 for all values of p and
f~~j (~
,
~.
1+
4’
4’
IQ1
/j~jn
U
Fig. 1. Pin arrangement with strongest pinning.
the maximum gradient of the orderparameter. Such strong pins are close to direct summation and we get from (1) 1 /, F ~j H2 LI2 pmax PI~o c 110 12~ ~
resu 0 re L J’ F 2~’4/5OO 2 ~3 IA\ p P J p1 C too, yields much to small 44C66 ~ values for F~.This discrep‘
L~3I
L
f~J
//
f~. 3. The correct summation in the weak pinning urnit was recently achieved by Larking and Ovchinnikov [8]. Their result may be interpreted in terms of fluxline “bundeling” [9] as the statistical summation of the N pinning forces in a finite coherence volume V~ = N/p (4irS3)R~2L~. This gives F~=f~..JN7V~.The length L~and the width R~of this volume are estimated as the distance over which (S2 >1/2 eq. (3), increases to the pinning force range, which is approximately the GL coherence length Unfortunately, the
H ~
—
‘-~
—
‘~
~
-
—
—
ancy experiments is probably the occurrence with of plastic deformation of the due FLLto[5,6] even for quite weak pinning. A qualitative treatment of plastic deformation appears quite difficult, and one may say that the summation of weak pinning forces is still an unsolved problem. In particular the linear dependence of F~on p, which seems to be well established experimentally [5] is not yet understood. On the other hand, the maximum possible loss-free current ofhard superconductors (2,c2 ~ 1) may well be estimated. The strongest possible pinning is ,
achieved by inclusions, precipitates, or voids of the shape and arrangement shown in fig. 1. The gaps along the FL direction suppress the movement of kinks, and the gaps perpendicular to the FLs prevent plastic flow around the obstacles.2 From fig. 1 we get the volume andthedensityp~sl/2Ld2. ofeachpin V~L~ 2/2d b/12 consists of pinThus the fraction ning material (b = Vp B/B~~ 2).Pinning arrangements with larger value of Vp are conceivable and might pin somewhat more effectively; an estimation of the pinning force is then rendered difficult by the strong perturbation. For our value of Vp it seems still 2)m~ fromreasonable Ginsburg—to estimatef~ ~‘I the gain (or loss) of Landau theory,Vu0H~ where (VI VuoH~is condensation energy and (VI ~PI2)m~(1 b)/~is
c2oincides with thatThe of ref. [4] ,but “scaling” of Ftl~eb-dependence H~HC’~near H~ dif2. The2is result ferent, F~ ir~a~c 1 b instead of (1 b) (5)may also be written as “~
—
—
—
Fpm~ MrevB/l 2~,
(6)
where Mrev = (H B/u 2 is the re0) the(H~2 H)/2g versible magnetization of superconductor. A similar expression, with the length 12~replaced by the FL spacing, was obtained by a crude estimation in ref. [1]. The maximum transversal critical current is/c ~Mrey/l2~ ~H~ 2~ ~Hc3/2/K’/2 for H<
—
—
The author wishes to acknowledge stimulating discussions with A.M. Campbell, J.E. Evetts, and E.J. Kramer. References [11 A.M. Campbell and J.E.
Evetts, Adv. in Phys. 90 (1972)
199.
485
Volume 77A, number 6
[21 R. Labusch, Crystall Lattice
PHYSICS LETTERS
Defects 1(1969)1. [3] K. Yamafuji and F. Irie, Phys. Lett. A 25 (1967) 387. [4] E.J. Kramer, J. Appl. Phys. 44 (1973) 1360. [5] E.J. Kramer, J. Appl. Phys. 49 (1978) 742. [61 A.M. Campbell, Phil. Mag. B 37 (1978) 149. [71 A.M. Campbell, J. de Physique 39 (1978) 66-619.
486
23 June 1980
[81 A.I. Larking and Yu.N. Ovchinnikov, J. Low Temp. Phys. 34 (1979) 409. [9] R. Schmucker and H. Kronmüller, Phys. Stat. Sol. (b) (1974) 181. [101 R.G. Boyd, Phys. Rev. 145 (1966) 255.
PHYSICS LETTERS
23 June 1980
THE MAXIMUM POSSIBLE LOSS-FREE CURRENT OF TYPE-lI SUPERCONDUCTORS WITH FLUX PINNING E.H. BRANDT Max-Planck-Institut fiir Meraliforsehung, Inst itut ftir Physik, Stuttgart, Germany Received 27 March 1980
The “dilute limit” and the “pinning threshold” are shown not to exist. An upper limit for the critical current in irreversible type-LI superconductors is given.
The maximum loss-free current density, ~ a typeII superconductor can support due to pinning of flux lines at material inhomogeneities is as important a material property as the transition temperature and the upper critical field H~2(O).Although there exists a large number of experimental and theoretical papers on/a [1] we are far from an understanding of the observed critical currents even in cases where the metallurgical structure of the superconductor is known. The problem is not so much the elementary pinning forces, which in many cases can be estimated satisfactorily from the Ginsburg—Landau and BCS theories, It is rather the summation of these forces to a volume pinning force F~= JcB. There are essentially three different methods, forp the specialsummation case of identical pointwhich pins we withdiscuss density and with maximum pinning force f~. 1. If the flux-line lattice (FLL) is soft it can adjust to the pins and each pin exerts its maximum force.
uli of the FLL. As discussed in detail by Kramer [5] and Campbell [6] the dilute limit formula does not explain the experiments. Even if the threshold is ignored (this was done in many papers) and even if the formula is improved to account for the periodicity of pinning forces due to the periodic FLL, the theoretical values for F~are still too small by several orders of magnitude. In fact the experimental values are closer to the direct summation (1). There are various reasons for the failure of formula (1). The most important one is that the “dilute limit” does not exist. Dilute means that, at the position of one pin, the flux-line displacements s caused by other pins may be neglected. This condition is never satis2) infied for linearly large specimens the mean square creases with the since specimen diameter D. (s This fact is due to the slow spatial decrease of the displacement field caused by one pin. One has [81 (f2> IC
2 +y2) + z2)\ 1/2 (3)
44
This is the case of direct summation, F —pf. P
(Is(r) —s(O)I 2> = (1)
p
2. The summation formula applied most often was derived in the “dilute limit” (p 0 but still a large number of pins) [2—41: F = 0 f ~fti~ r (2) -÷
p
thr’
p
>
thr
where fthr d2(c,~c 1/2is a threshold force, d is 66) c the flux-line spacing, and 44 and c66 are elastic mod484
(x
C “ 44C66
C66
2
2
where r = (x;y; z) and (f ) for periodic pinfling forces. Thus, if the FLL is shifted across a sufficiently large array of dilute pins the FL displacements at a given pin~‘deven oscillate for statistically with ampli2>1!2 weak pins. Thislarge means there tude is no (sthreshold. The divergence of (s ) in the limit D -÷°° was removed in ref. [2] by introduction of the Labusch parameter ~L’ which leads to an exponential decrease of s at larger distances. However, one can show that with-
Volume 77A, number 6
PHYSICS LETTERS
23 June 1980
in Labusch’s theory, below the threshold and for arbitrary shape of the pinning potential Unot only does FP=(—VU>vanishbutalsotheparameteraL=(V2U) (iF the averaging is performed properly [7]). With the consistent value = 0 inserted, various expressions in ref. [2] diverge and loose their sense. A correct theory should give aL ~ 0 and F~ 0 for all values of p and
f~~j (~
,
~.
1+
4’
4’
IQ1
/j~jn
U
Fig. 1. Pin arrangement with strongest pinning.
the maximum gradient of the orderparameter. Such strong pins are close to direct summation and we get from (1) 1 /, F ~j H2 LI2 pmax PI~o c 110 12~ ~
resu 0 re L J’ F 2~’4/5OO 2 ~3 IA\ p P J p1 C too, yields much to small 44C66 ~ values for F~.This discrep‘
L~3I
L
f~J
//
f~. 3. The correct summation in the weak pinning urnit was recently achieved by Larking and Ovchinnikov [8]. Their result may be interpreted in terms of fluxline “bundeling” [9] as the statistical summation of the N pinning forces in a finite coherence volume V~ = N/p (4irS3)R~2L~. This gives F~=f~..JN7V~.The length L~and the width R~of this volume are estimated as the distance over which (S2 >1/2 eq. (3), increases to the pinning force range, which is approximately the GL coherence length Unfortunately, the
H ~
—
‘-~
—
‘~
~
-
—
—
ancy experiments is probably the occurrence with of plastic deformation of the due FLLto[5,6] even for quite weak pinning. A qualitative treatment of plastic deformation appears quite difficult, and one may say that the summation of weak pinning forces is still an unsolved problem. In particular the linear dependence of F~on p, which seems to be well established experimentally [5] is not yet understood. On the other hand, the maximum possible loss-free current ofhard superconductors (2,c2 ~ 1) may well be estimated. The strongest possible pinning is ,
achieved by inclusions, precipitates, or voids of the shape and arrangement shown in fig. 1. The gaps along the FL direction suppress the movement of kinks, and the gaps perpendicular to the FLs prevent plastic flow around the obstacles.2 From fig. 1 we get the volume andthedensityp~sl/2Ld2. ofeachpin V~L~ 2/2d b/12 consists of pinThus the fraction ning material (b = Vp B/B~~ 2).Pinning arrangements with larger value of Vp are conceivable and might pin somewhat more effectively; an estimation of the pinning force is then rendered difficult by the strong perturbation. For our value of Vp it seems still 2)m~ fromreasonable Ginsburg—to estimatef~ ~‘I the gain (or loss) of Landau theory,Vu0H~ where (VI VuoH~is condensation energy and (VI ~PI2)m~(1 b)/~is
c2oincides with thatThe of ref. [4] ,but “scaling” of Ftl~eb-dependence H~HC’~near H~ dif2. The2is result ferent, F~ ir~a~c 1 b instead of (1 b) (5)may also be written as “~
—
—
—
Fpm~ MrevB/l 2~,
(6)
where Mrev = (H B/u 2 is the re0) the(H~2 H)/2g versible magnetization of superconductor. A similar expression, with the length 12~replaced by the FL spacing, was obtained by a crude estimation in ref. [1]. The maximum transversal critical current is/c ~Mrey/l2~ ~H~ 2~ ~Hc3/2/K’/2 for H<
—
—
The author wishes to acknowledge stimulating discussions with A.M. Campbell, J.E. Evetts, and E.J. Kramer. References [11 A.M. Campbell and J.E.
Evetts, Adv. in Phys. 90 (1972)
199.
485
Volume 77A, number 6
[21 R. Labusch, Crystall Lattice
PHYSICS LETTERS
Defects 1(1969)1. [3] K. Yamafuji and F. Irie, Phys. Lett. A 25 (1967) 387. [4] E.J. Kramer, J. Appl. Phys. 44 (1973) 1360. [5] E.J. Kramer, J. Appl. Phys. 49 (1978) 742. [61 A.M. Campbell, Phil. Mag. B 37 (1978) 149. [71 A.M. Campbell, J. de Physique 39 (1978) 66-619.
486
23 June 1980
[81 A.I. Larking and Yu.N. Ovchinnikov, J. Low Temp. Phys. 34 (1979) 409. [9] R. Schmucker and H. Kronmüller, Phys. Stat. Sol. (b) (1974) 181. [101 R.G. Boyd, Phys. Rev. 145 (1966) 255.