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Slow magnetic flux diffusion in type-II superconductors
Solid State Communications, Vol. 103, No. 7, pp, 399-401. 1997 Q 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038- 1098l97 $17X0+.00
PII: SOO38-1098(97)00190-7
SLOW MAGNETIC FLUX DIFFUSION IN TYPE-II SUPERCONDUCTORS F. Bass, B.Ya. Shapiro* and M. Shvartser Institute of Superconductivity, Department of Physics, Bar-Ilan University, Ramat Can 52900, Israel (Received 7 April 1997; accepted 28 April 1997 by S. Alexander)
The nonlinear dynamics of magnetic flux, moving in a vortex-free superconductor is considered. It has been shown that the time dependent characteristic size R of magnetic flux domains is described by the scaling behavior R - f, where Y= l/4 for circular droplets and Y= l/3 for elongated asymmetric droplets. These domains possess a sharp magnetic flux front and exist for a macroscopic long time in contradiction with usual diffusive behavior. 0 1997 Published by Elsevier Science Ltd Keywords: A. superconductors.
1. INTRODUCTION The dynamics of magnetic flux in type II superconductors has attracted the attention of many research groups. Since the penetration of magnetic flux is of fundamental importance for the superconducting properties of a sample, it has been studied in numerous investigations, in particular in conventional type II superconductors [ 11. Studies of patterns in the magnetic flux distribution in superconductors have generally been restricted to equilibrium configurations such as those observed in the intermediate state while out of equilibrium distributions are usually described in terms of Bean [2] or other kinds of critical state models. In these static models the magnetic front between the magnetic flux state and the Meissner superconducting state is considered to be a flat plane. However, since these flux distribution do not correspond to an equilibrium distribution, the system can eventually become unstable and flux enters the sample in momentary events in the form of flux droplets [3] and dynamics of the flux becomes relevant, These dynamics determine all flux features including heat release, shape of the magnetic flux front and the existence of large, long-lived flux drops surrounded by the Meissner phase. It seems at first glance that the magnetic flux diffusion must result in a very fast destruction of these states, In the present paper we describe this flux dynamics in detail and show, in
particular, that the flux motion is described by some scaling universal behavior, manifested in a sharp flux front and anomalously slow magnetic flux diffusion. 2. BASIC EQUATIONS In the vortex liquid hydrodynamics approximation in the flux flow regime the electric field E(r, t) induced by the moving vortices is related with the local electric current J(r, t) by the nonlinear Ohm law E(r, t) = p(B)J(r, t) where B is the modulus of the magnetic induction B&y) and p(B) can be identified as a field dependent resistivity in its convenient Bardeen-Stephen form [4]:
Bvo
p(B) = 47rrpocuBlc2= -. V
(1)
It is clear that the magnetic induction B(r, t) obeys the Maxwell equations which can be represented in the form dB dt=
- $7
X [p(B).(V X B)].
(2)
Here q(T) is the vortex viscosity, r = (x, y) and cpois a unit magnetic flux (for thick samples, B, = B, = 0). This equation can be considered as a nonlinear diffusion equation whose solution depends both on geometry and on the boundary conditions [5]. 3. PLANE FLUX FRONT
* Corresponding author.
If some magnetic flux + enters into a long superconducting strip its magnetic induction B can be taken to
SLOW MAGNETIC
400
FLUX DIFFUSION
depend only on the coordinate x perpendicular to the direction of the travelling flux and obeys the equation (3) B(x-w)
= 0,
where a = pea. This equation must be supplemented conservation law in the form +@Z
I
B(x, t) dx =
by the flux
a.
(4)
IN TYPE-II
Abrikosov vortices. Such droplets appear both as a result of internal flux front instability of the flat flux front [6] and as a consequence of magnetic flux penetration from the sample boundary 131. Magnetic induction of the flux inside the droplets obeys equation (2) which in cylindrical coordinates has the form a a aB(r, t) aB(r,t) = ;%rB(r, t>---at dr ’
-
(11)
where the boundary condition and the conservation of magnetic flux have the form
law
B(r-co)
0
= 0,
+UZ
Looking for a solution of equation
(3) in the form [5]
B(x, t) = P,t%),
r
27r
B(r, t)r dr = +.
(12)
(5) Looking for a solution of equation form
=X&t-m,
we immediately obtain from equations (4) and (5) the relation between m and k: m = k = l/3 and an equation for the function f(r)
$f!!L+ypO,
(6)
B(r,t) =
~S&s”),
decreasing
at x -
w (7)
I> 3-09
where the constants Pt. 02 in equation (5) are completely defined by the flux conservation law p, = (&J2)1’3,
p* = (Jlacp)1’3,
(13)
we obtain m = l/4, k = l/2 and an equation for function f as follows
$cf*)+
f$f
+f
=o,
(8)
f(l) = St: - t2>. Substituting fit) magnetic induction
(15) in equation
(13) we get for the
(5: - r2/&)/8poa!$,
r*/$s
.$,
r2/Ji
2 .$i,
B(r, t) =
(16) 1 0,
where ,$ = 4dw. The droplet front described by equation with the time dependent velocity
dro
The solution then reads $t -
“3(g -
x2/3aP3),
x213aP3 5 s;, x2/3atu3 2 g.
(9) The sharp flux front described by equations and (9) moves with the time dependent velocity “Z
&O
-= df
a9
If3t-2l3
(> 3
OF MAGNETIC
H
‘J4t_
3,4
(17)
r
Similar to the plane flux front, the droplet boundary also obeys a nonlinear diffusion law r(t) - t”4 much slower than usual diffusion or a diffusion of a plane flux front. 5. FAST DIFFUSION
where x0 is defined from the condition r = ro, The flux front coordinate x0(t) - (LZ@)“~Z”~ changes slower than is expected for usual diffusion, when x0(t) - t”*. 4. DROPLET
@vow
(16) moves
(8)
(10)
’
2
V=dt=
B(x, t) = 0,
(14)
which has the exact solution
{
f = 0,
(11) with scaling
.$= rIC2tm,
C,t-kf(.9,
;$cf*)+
which has the evident solution, f=
Vol. 103, No. 7
SUPERCONDUCTORS
FLUX
In most experimental situations, magnetic flux forms large droplets containing hundreds and even thousands of
OF FLUX DROPLETS
If the flux droplet enters a superconductor containing homogeneously distributed flux B,,, its dynamics is very different from those in flux free samples. Indeed, in this case a magnetic flux B inside the droplet can be considered as a sum of the background field B. and deviation b(r, t) and one may rewrite equation (11) in the form
a&, t) -= at
~~~~~
a
r
ar
--r
Wv)
-+
dr
cpoa a
7
,rb(r,t)---.
at-Q,t) ar (18)
Vol. 103, No. 7 SLOW MAGNETIC FLUX DIFFUSION IN TYPE-II SUPERCONDUCTORS
401
Let us consider the limit b e B,, related to the last stage of the droplet dispersion inside the sample. In this case equation (18) turns into a well-known diffusion equation
magnetic flux areas stretched along the strip. Their characteristic size in the direction of flux expansion changes as - t1’3 in the last, third, stage of the flux penetration. Let us estimate some characteristic times for which a al+-, t) --r-----_ -= cpoarBo a ab(r’ ‘) p+xBoAb(r, t), (19) flux spot reaches a radius R. In particular for the flux r ar ar at droplet, possessing magnetic flux Cp= 102po and radius with the diffusion coefficient Kg = p&lo. The solution R = 1 cm and parameter (Y= lo5 cm2 din-’ s-’ we is immediately obtain from, equation (17), St - lo7 s which is a macroscopically large time. At the same (20) time the characteristic time of the linear diffusion is estimated as 6t - R2/~0aBo. Substituting B. = lo2 Gs, where c is the magnetic field in the spot center at some R = 1 cm, one receives 6t - 1 s which is much faster than the nonlinear diffusion stage. moment of time to multiplied by to: c = B(0, t)tO. In this case the sharp boundary of the droplet does not work was supported by the exist and its characteristic size changes as 4, meaning Acknowledgements-This Israel Ministry of Sciences and Arts, German-Israel essentially diffusive behavior. Foundation and Israel Academy of Sciences. We also would like to thank the Bar-Ban Minerva Center for 6. CONCLUSION Superconductivity for permanent support. Thus, one can distinguish a few different manifestations of the magnetic flux droplets dynamics in superconductors. First of all the magnetic flux can form droplets of rather small size R which very slowly changes with time R - t1’4. The vortex-free space between magnetic flux droplets manifests itself as long-lived “Meissner spots” which do not overlap due to the flux front’s sharpness. Secondly, the droplets can overlap and the ‘ ‘Meissner spots’ ’ are filled by the flux, which expands according to the linear diffusion mechanism (R-t’“). These coagulated droplets form large,
REFERENCES 1. Huebner, R.P., Magnetic Flux Structures in Superconductors. Springer, Berlin, 1979. 2. Bean, C.P., Rev. Mod. Phys., 36, 1964, 31. 3. Aranson, I.B., Shapiro, B.Ya. and Vinokur, V., Phys. Rev. L&t., 76, 1996, 142.
4. Bardeen, J. and Stephen, M.J., Phys. Rev., 140, 1965, Al 197. 5. Landau, L.D. and Lifshits, E.M., Hydrodynamics. Pergamon Press, Oxford, 199 1. 6. Leiderer, P. et al., Phys. Rev. L&t., 71, 1993,2646.
PII: SOO38-1098(97)00190-7
SLOW MAGNETIC FLUX DIFFUSION IN TYPE-II SUPERCONDUCTORS F. Bass, B.Ya. Shapiro* and M. Shvartser Institute of Superconductivity, Department of Physics, Bar-Ilan University, Ramat Can 52900, Israel (Received 7 April 1997; accepted 28 April 1997 by S. Alexander)
The nonlinear dynamics of magnetic flux, moving in a vortex-free superconductor is considered. It has been shown that the time dependent characteristic size R of magnetic flux domains is described by the scaling behavior R - f, where Y= l/4 for circular droplets and Y= l/3 for elongated asymmetric droplets. These domains possess a sharp magnetic flux front and exist for a macroscopic long time in contradiction with usual diffusive behavior. 0 1997 Published by Elsevier Science Ltd Keywords: A. superconductors.
1. INTRODUCTION The dynamics of magnetic flux in type II superconductors has attracted the attention of many research groups. Since the penetration of magnetic flux is of fundamental importance for the superconducting properties of a sample, it has been studied in numerous investigations, in particular in conventional type II superconductors [ 11. Studies of patterns in the magnetic flux distribution in superconductors have generally been restricted to equilibrium configurations such as those observed in the intermediate state while out of equilibrium distributions are usually described in terms of Bean [2] or other kinds of critical state models. In these static models the magnetic front between the magnetic flux state and the Meissner superconducting state is considered to be a flat plane. However, since these flux distribution do not correspond to an equilibrium distribution, the system can eventually become unstable and flux enters the sample in momentary events in the form of flux droplets [3] and dynamics of the flux becomes relevant, These dynamics determine all flux features including heat release, shape of the magnetic flux front and the existence of large, long-lived flux drops surrounded by the Meissner phase. It seems at first glance that the magnetic flux diffusion must result in a very fast destruction of these states, In the present paper we describe this flux dynamics in detail and show, in
particular, that the flux motion is described by some scaling universal behavior, manifested in a sharp flux front and anomalously slow magnetic flux diffusion. 2. BASIC EQUATIONS In the vortex liquid hydrodynamics approximation in the flux flow regime the electric field E(r, t) induced by the moving vortices is related with the local electric current J(r, t) by the nonlinear Ohm law E(r, t) = p(B)J(r, t) where B is the modulus of the magnetic induction B&y) and p(B) can be identified as a field dependent resistivity in its convenient Bardeen-Stephen form [4]:
Bvo
p(B) = 47rrpocuBlc2= -. V
(1)
It is clear that the magnetic induction B(r, t) obeys the Maxwell equations which can be represented in the form dB dt=
- $7
X [p(B).(V X B)].
(2)
Here q(T) is the vortex viscosity, r = (x, y) and cpois a unit magnetic flux (for thick samples, B, = B, = 0). This equation can be considered as a nonlinear diffusion equation whose solution depends both on geometry and on the boundary conditions [5]. 3. PLANE FLUX FRONT
* Corresponding author.
If some magnetic flux + enters into a long superconducting strip its magnetic induction B can be taken to
SLOW MAGNETIC
400
FLUX DIFFUSION
depend only on the coordinate x perpendicular to the direction of the travelling flux and obeys the equation (3) B(x-w)
= 0,
where a = pea. This equation must be supplemented conservation law in the form +@Z
I
B(x, t) dx =
by the flux
a.
(4)
IN TYPE-II
Abrikosov vortices. Such droplets appear both as a result of internal flux front instability of the flat flux front [6] and as a consequence of magnetic flux penetration from the sample boundary 131. Magnetic induction of the flux inside the droplets obeys equation (2) which in cylindrical coordinates has the form a a aB(r, t) aB(r,t) = ;%rB(r, t>---at dr ’
-
(11)
where the boundary condition and the conservation of magnetic flux have the form
law
B(r-co)
0
= 0,
+UZ
Looking for a solution of equation
(3) in the form [5]
B(x, t) = P,t%),
r
27r
B(r, t)r dr = +.
(12)
(5) Looking for a solution of equation form
=X&t-m,
we immediately obtain from equations (4) and (5) the relation between m and k: m = k = l/3 and an equation for the function f(r)
$f!!L+ypO,
(6)
B(r,t) =
~S&s”),
decreasing
at x -
w (7)
I> 3-09
where the constants Pt. 02 in equation (5) are completely defined by the flux conservation law p, = (&J2)1’3,
p* = (Jlacp)1’3,
(13)
we obtain m = l/4, k = l/2 and an equation for function f as follows
$cf*)+
f$f
+f
=o,
(8)
f(l) = St: - t2>. Substituting fit) magnetic induction
(15) in equation
(13) we get for the
(5: - r2/&)/8poa!$,
r*/$s
.$,
r2/Ji
2 .$i,
B(r, t) =
(16) 1 0,
where ,$ = 4dw. The droplet front described by equation with the time dependent velocity
dro
The solution then reads $t -
“3(g -
x2/3aP3),
x213aP3 5 s;, x2/3atu3 2 g.
(9) The sharp flux front described by equations and (9) moves with the time dependent velocity “Z
&O
-= df
a9
If3t-2l3
(> 3
OF MAGNETIC
H
‘J4t_
3,4
(17)
r
Similar to the plane flux front, the droplet boundary also obeys a nonlinear diffusion law r(t) - t”4 much slower than usual diffusion or a diffusion of a plane flux front. 5. FAST DIFFUSION
where x0 is defined from the condition r = ro, The flux front coordinate x0(t) - (LZ@)“~Z”~ changes slower than is expected for usual diffusion, when x0(t) - t”*. 4. DROPLET
@vow
(16) moves
(8)
(10)
’
2
V=dt=
B(x, t) = 0,
(14)
which has the exact solution
{
f = 0,
(11) with scaling
.$= rIC2tm,
C,t-kf(.9,
;$cf*)+
which has the evident solution, f=
Vol. 103, No. 7
SUPERCONDUCTORS
FLUX
In most experimental situations, magnetic flux forms large droplets containing hundreds and even thousands of
OF FLUX DROPLETS
If the flux droplet enters a superconductor containing homogeneously distributed flux B,,, its dynamics is very different from those in flux free samples. Indeed, in this case a magnetic flux B inside the droplet can be considered as a sum of the background field B. and deviation b(r, t) and one may rewrite equation (11) in the form
a&, t) -= at
~~~~~
a
r
ar
--r
Wv)
-+
dr
cpoa a
7
,rb(r,t)---.
at-Q,t) ar (18)
Vol. 103, No. 7 SLOW MAGNETIC FLUX DIFFUSION IN TYPE-II SUPERCONDUCTORS
401
Let us consider the limit b e B,, related to the last stage of the droplet dispersion inside the sample. In this case equation (18) turns into a well-known diffusion equation
magnetic flux areas stretched along the strip. Their characteristic size in the direction of flux expansion changes as - t1’3 in the last, third, stage of the flux penetration. Let us estimate some characteristic times for which a al+-, t) --r-----_ -= cpoarBo a ab(r’ ‘) p+xBoAb(r, t), (19) flux spot reaches a radius R. In particular for the flux r ar ar at droplet, possessing magnetic flux Cp= 102po and radius with the diffusion coefficient Kg = p&lo. The solution R = 1 cm and parameter (Y= lo5 cm2 din-’ s-’ we is immediately obtain from, equation (17), St - lo7 s which is a macroscopically large time. At the same (20) time the characteristic time of the linear diffusion is estimated as 6t - R2/~0aBo. Substituting B. = lo2 Gs, where c is the magnetic field in the spot center at some R = 1 cm, one receives 6t - 1 s which is much faster than the nonlinear diffusion stage. moment of time to multiplied by to: c = B(0, t)tO. In this case the sharp boundary of the droplet does not work was supported by the exist and its characteristic size changes as 4, meaning Acknowledgements-This Israel Ministry of Sciences and Arts, German-Israel essentially diffusive behavior. Foundation and Israel Academy of Sciences. We also would like to thank the Bar-Ban Minerva Center for 6. CONCLUSION Superconductivity for permanent support. Thus, one can distinguish a few different manifestations of the magnetic flux droplets dynamics in superconductors. First of all the magnetic flux can form droplets of rather small size R which very slowly changes with time R - t1’4. The vortex-free space between magnetic flux droplets manifests itself as long-lived “Meissner spots” which do not overlap due to the flux front’s sharpness. Secondly, the droplets can overlap and the ‘ ‘Meissner spots’ ’ are filled by the flux, which expands according to the linear diffusion mechanism (R-t’“). These coagulated droplets form large,
REFERENCES 1. Huebner, R.P., Magnetic Flux Structures in Superconductors. Springer, Berlin, 1979. 2. Bean, C.P., Rev. Mod. Phys., 36, 1964, 31. 3. Aranson, I.B., Shapiro, B.Ya. and Vinokur, V., Phys. Rev. L&t., 76, 1996, 142.
4. Bardeen, J. and Stephen, M.J., Phys. Rev., 140, 1965, Al 197. 5. Landau, L.D. and Lifshits, E.M., Hydrodynamics. Pergamon Press, Oxford, 199 1. 6. Leiderer, P. et al., Phys. Rev. L&t., 71, 1993,2646.