VOLUME 80, NUMBER 11
PHYSICAL REVIEW LETTERS
16 MARCH 1998
Corrugation Flux Front Instability in Type-II Superconductors
F. Bass, B. Ya. Shapiro, and M. Shvartser
Institute of Superconductivity, Department of Physics, Bar-Ilan University, Ramat Gan 52900, Israel
(Received 16 June 1997)
The dynamics of vortex flux entering a superconductor containing immovable flux of the opposite sign
(antivortices) is considered. We obtain a solution for a flat moving interface separating the positive and
negative flux areas. When the velocity of the positive flux exceeds some critical value, a corrugation
instability of the interface develops. [S0031-9007(98)05548-3]
PACS numbers: 74.60.Ge
The study of magnetic flux penetration into type-II superconducting samples has attracted the attention of many
research groups [1]. This interest is motivated by the fact
that this phenomenon can be viewed as a prototype of a
general class of macroscopic nonlinear dynamic phenomena, as well as by its importance for applications.
Traditionally, the magnetic front between the magnetic
flux state and the Meissner superconducting state has
been considered to be a flat plane [2]. However, recent magneto-optical experiments [3–6] demonstrate that
nonuniform flux penetration occurs, in particular, upon
exposing a previously magnetized sample to a weak magnetic field of the opposite sign. While at low temperatures, and flux (aligned along the new field) gradually
enters the sample and the flux boundary remains flat; the
situation changes dramatically at higher temperatures.
In particular, in a YBa2 Cu3 O7 bulk superconductor,
at temperatures above 47 K, the boundary between the
distinct flux areas is strongly perturbed and a magnetic
field, of magnitude 120 G, penetrates the sample through
a sudden nucleation of magnetic domains [3–5]. This
instability exists only in a relatively narrow temperature
“window” and has not been observed above 80 K. In
superconducting Nb, the structure and growth of the
magnetic domain appear similar to that seen for a viscousfingering growth phenomena in solid-liquid systems [7].
The temperature window for the instability in this sample
stretches at least from 3 to 7 K [6].
In this paper we report on flux-antiflux dynamics
in type-H superconductors. In particular, we find a
solution for a flat moving interface separating positive and
negative flux areas in a superconductor. We predict that
this interface becomes unstable when the flux velocity
exceeds some critical value due to the “overheating”
of the interface caused by vortex-antivortex annihilation.
The growth of the instability gives rise to a nonlinear
pattern whose nature remains to be determined.
Let us consider the situation when vortex flux enters
a sample containing immovable flux of the opposite sign
(antivortices). We start with a model of a two-component
vortex gas [8] spatially homogeneous along the z-axis,
which is valid for the experimentally interesting situation
of low magnetic fields. Here, the typical spacing between
0031-9007y98y80(11)y2441(4)$15.00
vortices a0 greatly exceeds the vortex-vortex (or vortexantivortex) interaction radius j. The vortex-vortex repulsion, since it conserves the number of vortices, does not
play a significant role, and will be ignored.
In this approximation, the vortex-antivortex dynamics
obeys the well-known equations of recombination theory
[8,9]
n1 n2
≠n1
!
yd 1
1 = ? sn1 !
­ 0,
(1)
≠t
t
n1 n2
≠n2
1
­ 0,
(2)
≠t
t
where n1 and n2 are the vortex and antivortex densities,
respectively; t 21 ­ yj is the recombination rate; j is the
annihilation cross-section, which is of the order of the co!
is the vortex
herence length of the superconductor; and y
!
velocity. In the creep regime, y is strongly temperature
dependent
!
!
­y
(3)
y
FF exps2UyT d ,
!
where U is the pinning potential; and y
FF is the flux
velocity in the flux-flow regime.
The set of Eqs. (1) and (2) must be completed by the
heat transfer equation in the form:
dQ
≠T
(4)
­ k=2 T 1
2 gsT 2 T0 d ,
≠t
dt
where
dQ
n1 n2
(5)
­
Q0
dt
cp t
is determined by the energy released by vortex-antivortex
annihilation. Here k is the heat coefficient, cp is the
heat capacity, T0 is the sample temperature in the absence
of flux motion, Q0 ­ f0 Hc1 y2p is the heat released by
annihilation of a single vortex-antivortex pair, Hc1 is the
lower critical field, f0 is unit flux, and g 21 ­ tR is the
characteristic time of temperature relaxation.
The set of Eqs. [(1)–(4)] completed by the boundary
conditions describes all features of the model. This hydrodynamical approach is correct if we consider variations
on a scale much larger when a0 .
We start from the plane flux front dynamics when
all of the functions described by Eqs. [(1)–(4)] depend
© 1998 The American Physical Society
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VOLUME 80, NUMBER 11
PHYSICAL REVIEW LETTERS
only on the spatial coordinate parallel to the flux-front
propagation. In this case the problem becomes one
dimensional, as described by the set of equations:
n1 n2
≠n1
≠
1
sn1 yd 1
­ 0;
≠t
≠x
t
(6)
n1 n2
≠n2
1
­ 0,
≠t
t
(7)
≠2 T
≠T
n1 n2
­k 2 1
Q0 2 gsT 2 T0 d ,
≠t
≠x
cp t
(8)
and the boundary conditions n1 sx ! 2`d ­ n2 sx !
`d ­ n0 .
Working to the leading order in the small parameter
dTyT0 , where dT is the temperature growth caused by
the vortex-antivortex annihilations during the effective
heating time (which cannot exceed the relaxation time tR ),
we immediately obtain from Eq. (8) that T ­ T0 . The
maximal shift of the temperature dT may be estimated
as dT . Q0 n0 tR ytA cp where tA ­ t0 yn0 is the vortexantivortex annihilation time.
In this approximation, and neglecting magnetic screening effect, the pair of nonlinear equations (6) and (7) may
be solved exactly. In particular, substituting for n2 from
Eq. (7) in Eq. (6) and introducing a new variable
Z t
t 21
n1 sx, t 0 d dt 0 ­ usx, td ,
(9)
z ­x2
n1 sz d ­ n0
1
;
1 1 exps2n0 z yy0 td
n2 sz d ­ n0
exps2n0 z yy0 td
,
1 1 exps2n0 z yy0 td
n2 sz , y, td ­ n2 sz d 1 Qsz , y, td;
T ­ T0 sz d 1 D
µ
∂
Dsz , y, td
;
y ­ y0 1 1
T0
µ
∂
Dsz , y, td
21
21
t ­ t0 1 1
.
T0
(10)
(11)
y0 ; ysT0 d .
(13)
(14)
Here we neglect the influence of temperature fluctuations on the heat capacity cp and on the relaxation coefficient g, as this does not affect the physical picture. (We
implied that U ­ T0 .)
Substituting these relations in the initial set of Eqs. (1),
(2), (4) we obtain after linearization the following set of
equations for the deviations:
∂
µ 2
≠2 D
≠D
y0 ≠D
≠ D
1
2 gD
2
­k
≠t
2 ≠z
≠y 2
≠z 2
1 Q0
(12)
n1 sz , y, td ­ n1 sz d 1 Csz , y, td;
one can transform the pair of equations (6) and (7) to the
following single equation:
Performing an integration over time and solving the
equation obtained by the method of characteristics, we
obtain the traveling interface solution:
y0
t
2
It is interesting to note that the vortex-antivortex
interface moves into the antivortex-filled domain with
the velocity y0 y2, the average of the velocity y0 of the
moving vortices and the zero velocity of the antivortices.
The spatial distributions of vortex and antivortex flux
densities overlap, forming an interface region where
the vortices of opposite signs coexist. The scale of
characteristic width of this region DL may be estimated
from Eqs. (11) and (12) as DL . ytyn0 , sn0 jd21 ,
a02 yj. This is much larger than the intervortex distance
a0 , so that there is a macroscopically large area in
which the total magnetic induction B ­ f0 sn2 2 n1 d ­
n0 f0 tanhsn0 z yy0 td is suppressed.
The vortex velocity sy ~ ,Bd approaches a constant at
the interface point where B ! 0, justifying our considerations. In particular, this implies that our solution is valid
in the interface region even in the presence of magnetic
screening [10].
Let us consider the stability of this vortex-antivortex
interface with respect to small deviations from its initial
plane shape. In this case one can look for solutions of
Eqs. (1), (2), (4) in the form
2`
≠2 u
≠2 u
n0 ≠ exps2ud
1
y
2
­ 0.
≠t 2
≠x≠t
t
≠t
16 MARCH 1998
fn 1 sz dQ 1 n2 sz dC 1 n1 sz dn2 sz d sDyT0dg
,
t0 cp
(15)
(16)
µ
∂
≠C
≠D
≠D
≠C
y0 ≠C
n1 sz d
fn 1 sz dQ 1 n2 sz dC 2 n1 sz dn2 sz d sDyT0 dg
­ 0,
y0
1
1
1 y1
1 y1
1
≠t
2 ≠z
T0
≠z
≠y
≠y
t0
(17)
y0 ≠Q
fn1 sz dQ 1 n2 sz dC 1 n1 sz dn2 sz d sDyT0 dg
≠Q
­ 0,
2
1
≠t
2 ≠z
t0
2442
(18)
VOLUME 80, NUMBER 11
PHYSICAL REVIEW LETTERS
!
where we assume that the vortex velocity y
also possesses
a component parallel to the interface yy sT d, where yy ø
y1 f1 1 Dsz , y, tdyT0 g, and y1 ø y0 .
It is evident that all of the preparations far from the
flux-antiflux interface cannot result in a front instability.
Therefore, one can assume that these perturbations must
be localized on the boundary. Looking for a solution in
the form
0 1
0
1
C
Csz , y, t
B C
B
C
@ Qsz , y, td A ­ expsvt 1 iky y 2 az 2 y2d @ Q A ,
D
Dsz , y, td
(19)
we obtain from Eqs. [(15)–(18)] after integration on z
a set of equations, and the requirement for the selfconsistency of these leads to the following equation for
the dimensionless growth rate: V ­ vt0 yn0
V 3 1 V 2 fsiq 1 1d 1 Aq2 1 g p 2 P p g 1
∑
∏
iq
Pp
V
1 sAq2 2 P p 1 g p d siq 1 1d 1 iqP p 1
1
2
2
iq
(20)
sAq2 1 g p d ­ 0 ,
2
where
q­
ky y1 t0
;
n0
t0
g ­g ;
n0
p
A­
kn0
;
y12 t0
Q0 n02
P­
.
4t0 T0 cp
Pp ­ P
t0
;
n0
(21)
Representing V in the form
V ­ V1 1 iV2 ,
(22)
we obtain for V1 in theplimit of g p ø 1y4A
p ø 1sq ! 0d
(since tA ytR ø 1 and kytA ¿ y1 ¿ kytR ):
∂
µ
1 Pp
1
2 1 q2 2
q4 .
(23)
V1 ­
p2
4 2g
16g p2
A positive value of V1 implies an instability, and this
occurs if P p . 2g p2 .
In dimensional variables [see Eq. (21)] this condition
reads
s
8T0 cp g 2
y0 $ yc ­
.
(24)
Q0 n03 j 2
For velocities of the flux exceeding this critical value yc
there is a band of unstable wave vectors, and a critical
length scale defined by the wave vector qc of the fastest
growing mode:
∂
µ p
P
2
p2
qc ­ 2g
21 .
(25)
2g p2
16 MARCH 1998
In dimensional variables we get sy0 $ yc d
y1 tR
lc ­ 2pykc . q
¿ a0 .
2
fy0 yyc2 2 1g
(26)
This is just the condition that we need in order to justify
using the hydrodynamical approach.
Thus, the dynamics of a type-II superconductor, containing vortex flux moving through immovable flux of the
opposite sign (anitvortices), strongly depends on the vortex velocity and on the temperature, which may be considered to be the main control parameter of the problem.
This dynamics possesses the following most important
properties:
(1) For a sufficiently small velocity of the vortices on
the interface y , yc the flux-antiflux interface propagates
with velocity yy2. This fact has an evident physical
explanation. Indeed, in the frame moving with velocity
yy2, vortices possess velocity yy2, while antivortices
have velocity 2yy2 and the flux-antiflux interface is
stationary.
(2) The characteristic width of the vortex-antivortex
coexistence region DL may be macroscopically large
DL , a02 yj ¿ a0 , where a0 and j are the intervortex
distance and linear cross sections of the vortex-antivortex
annihilation, respectively.
(3) If the vortex flux velocity exceeds its critical value
y0 . yc , then the flux-antiflux interface exhibits an instability. This instability is caused by the vortex-antivortex
overheating exceeding the thermal absorption in the sample. This type of instability is well known in plasma
physics as the “overheating instability” [11,12]. The critical velocity of this instability, determined by Eq. (24),
may be simply obtained in the following manner. The
temperature increase during the time tA of a single annihilation event may be estimated as Q0 n0 ycp tA [see Eq. (6)],
which must be multiplied by the number dN of the unit
events during the heat relaxation time tR , dN , tR ytA .
The relaxation caused by the sample lattice is the competing process in the system. The temperature relaxation
for time tR may be estimated as T0 ytR . It is evident that
the overheating instability arises under the condition
DQ
tR Q0 n0
T0
;
2
. 0,
tR
tA cp tA
tR
tR ­ g 21 ;
tA ­ sn0 yo jd21 ,
(27)
(28)
which is equivalent to the criterion defined by Eq. (24).
(4) The most rapidly growing mode of instability
occurs at wavelength lc . It seems reasonable to guess
that patterns emerging from this instability will, at least
initially, have a characteristic size of order lc ¿ a0 in
which case lc may be macroscopically large.
(5) The growth rate of the corrugation instability V1 sqd
shows that in the domain of instability the growing modes
develop for a wide range of wave vectors. The wave
length lm at which V1 vanishes, sets a length scale
2443
VOLUME 80, NUMBER 11
PHYSICAL REVIEW LETTERS
FIG. 1. The ratio y0 yyc vs temperature for Nb for which
Tc ­ 9 K. A corrugation instability arises only in a comparatively narrow temperature “window” of width DTW determined
by the inequality y0 sT0 d . yc sT0 d.
for the problem.
The set of Eqs. [(23)–(25)] tells us
p
that lc ­ 2lm , and the minimal wavelength of the
instability has the same order as the rapidly growing mode
lc . The existence of this length has a very clear physical
explanation. It appears as a balance between the excess
heat cp DQlm released in the fluctuation region of the
size of the order of lm for the time t $ tR and the heat
flow sy carried by the vortices and directed along the
interface, bringing this heat out from this hot domain due
to the temperature gradient. The balance relation gives
the condition of instability
DQlm . sy1 tR d2 T0 ylm ;
sy ­ 2cp sy12 tR d=y T ,
(29)
y12 tR
is the diffusion coefficient of the vortices
and
sliding along the interface and transmitting heat to the
lattice. On substituting DQ from Eq. (27) in Eq. (29) we
immediately obtain for lm the result of the linear theory
of instabilities defined by Eq. (26).
(6) The instability appears only in a relatively narrow
temperature window determined by the inequality y0 .
yc sT0 d. For instance, for an Nb sample with the following parameters: the Fermi energy eF ­ 3 3 104 K;
the Fermi momentum pF ­ 107 cm21 ; the super-
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16 MARCH 1998
conducting gap D0 ­ 13 K; the critical temperature
Tc ­ 9 K, the Debye frequency vD ­ 300 K; the
vortex viscosity h ­ 1027 dyn secycm2 ; the intervortex
distance a0 ­ 1024 cm; the annihilation cross section
j ­ 1025 cm; the Fermi velocity yF ­ 107 cmysec;
the ratio UyD0 ­ 1.2; the flux velocity in the flux-flow
regime yFF ­ 104 cmysec, where yFF , f0 Jc ych; Jc ­
cf0 n02 j ­ 105 Aycm and using for cp and g their dependence in the low temperature region cp ­ NseF dD0 sD0 y
T d3y2 exps2D0 yT d, where NseF d is the DOS at Fermi
2
level and g ­ ge-ph ­ T 3 yvD
, we conclude from
Eqs. (3) and (24) that the temperature window for flux
instability is very similar to that observed experimentally
[6] in this material, namely DTW ø 4 K, as can be seen
from Fig. 1.
We would like to thank Dr. M. Bazilevitch for helpful
discussions.
This work was supported by the Israel Ministry of
Sciences and Arts and German-Israel Foundation. We
would also like to thank the Bar-Ilan Minerva Center for
Superconductivity for permanent support.
[1] R. P. Huebner, Magnetic Flux Structures in Superconductors (Springer-Verlag, Berlin, 1979).
[2] C. P. Bean, Rev. Mod. Phys. 36, 31 (1964).
[3] V. K. Vlasko-Vlasov et al., Physica (Amsterdam) 222, 361
(1994).
[4] T. H. Johansen et al., in High Temperature Superconductors: Synthesis, Processing, and Large-Scale Applications,
edited by U. Balachandran, P. J. McGinn, and J. S. Abell
(The Minerals, Metals & Materials Society, Warrendale,
PA, 1996).
[5] M R. Koblichka et al., Europhs. Lett. (to be published).
[6] C. A. Duran, P. L. Gammel, R. E. Miller, and D. J. Bishop,
Phys. Rev. B 52, 75 (1995).
[7] J. S. Langer, Rev. Mod. Phys. 52, 1 (1980).
[8] V. V. Bryksin and S. N. Dorogovtsev, Sov. Phys. JETP 77,
791 (1993).
[9] E. S. Bakhanova et al., Sov. Phys. JETP 73, 1061 (1991).
[10] F. Bass, B. Ya Shapiro, I. Shapiro, and M. Shvartser,
Phys. Rev. B (to be published).
[11] J. Bok, Ann. Radioelectr. 15, 120 (1960).
[12] I. P. Shkarovsky, T. W. Jonston, and M. P. Bachinsky, The
Particle Kinetics of Plasmas (Addison-Wesley Publishing
Company, Reading, MA, 1966).