PHYSICAL REVIEW B
VOLUME 58, NUMBER 5
1 AUGUST 1998-I
Flux-antiflux interface in type-II superconductors
F. Bass and B. Ya. Shapiro*
Institute of Superconductivity, Department of Physics, Bar-Ilan University, Ramat Gan 52900, Israel
I. Shapiro
Faculty of Engineering, Tel-Aviv University, Tel-Aviv, 69978, Israel
M. Shvartser
Department of Physics, Bar-Ilan University, Ramat Gan 52900, Israel
~Received 15 December 1997; revised manuscript received 11 March 1998!
Dynamics of vortex flux entering the superconductor containing flux of the opposite sign ~antivortices! is
considered. The interface point separating flux-antiflux domains is described by the solution moving inside the
negative flux region. If vortices and antivortices penetrate a superconductor from the opposite sides of the
sample symmetrically, the stationary flux-antiflux spatial distribution is described by the universal scaling law.
Heat released by the vortex-antivortex annihilation forms a spatial temperature wave running across the
sample. When velocity of the flux exceeds some critical value, it is predicted that the overheating instability
will grow. @S0163-1829~98!05329-6#
I. INTRODUCTION
The dynamics of flux penetration of the superconductors
has for a long time1 attracted wide attention. It is well known
that magnetic flux penetrates pure type-II superconductors by
the Abrikosov vortices carrying a unit flux w 0 , whose penetration, motion, and spatial distribution were investigated in
detail. On the other hand, the situation when the vortices
interact in a superconductor with the flux of the opposite sign
attracts special interest. This situation appears in many experiments. In particular, it takes place when a dc bias current
creates vortices and antivortices on the opposite side of the
superconducting strip. Another phenomena now under intensive investigation arises upon exposing the previously magnetized sample to the weak magnetic field of an opposite
direction.
Studies of patterns in the magnetic flux distribution in
superconductors are attracting the attention of many research
groups2–5 whose magneto-optical experiments demonstrate
that nonuniform flux penetration occurs. At low temperatures, flux of opposite polarities gradually enter the sample
with the flat boundary between differently magnetized areas
resembling the front of the initial magnetization.
Unexpectedly, in a YBCO bulky superconductor, at temperatures above 47 K, the boundary between the distinct flux
areas is strongly perturbed and a reversed, 120 G of magnitude, magnetic field penetrates the sample through a sudden
nucleation of magnetic patterns.2–4 This instability exists
only in a, respectively, narrow temperature ‘‘window’’ and
has not been observed above 80 K. In the superconducting
Nb the structure and growth appear similar to that seen for a
viscous-fingering growth phenomena in solid-liquid
systems.6 The temperature ‘‘window’’ for instability in this
sample stretches at least from 3 to 7 K.5
There are two different possible origins for such dendritic
instabilities in superconductors. The first, vortex nucleation,
has already been considered.7 By analytical solving of the
0163-1829/98/58~5!/2878~8!/$15.00
PRB 58
Ginzburg-Landau equations for a very thin superconducting
strip it was found that giant normal spots appear at the strip
edges and evolve into single vortices immediately after creation. In the presence of random pinning centers, the resulting picture was very similar to the one observed experimentally in the magnetic-flux distribution. However, the
characteristic times of the discovered instabilities were of the
orders of microseconds, i.e., orders of magnitude faster than
the slow evolutions seen in the magneto-optic experiments.
Another possibility is related to the macroscopic instability of a moving flux-antiflux boundary. In this scenario the
dynamics of the magnetic flux in type-II superconductors is
described by diffusionlike equations for a viscous liquid
flow. In general, as is well known from hydrodynamics, the
viscous flow manifests a turbulent instability at sufficiently
high flow velocities.8 In particular, the magnetic-flux instability develops in a moving flux front when its movement is
associated with heat release. This instability is caused by the
imbalance between the heat released by the vortex-antivortex
annihilation and its relaxation ~overheating instability!.9
In this paper we report on flux-antiflux dynamics in
type-II superconductors. We calculate spatial distribution of
the flux-antiflux densities and of the temperature across the
sample in various experimental situations. We also consider
flux-antiflux interface instability that develops when the relative vortex-antivortex velocity exceeds some critical value.
II. BASIC EQUATIONS
We start with a model of two-component vortex gas10
spatially homogeneous along the z axes, which is valid for
the experimentally interesting situation of the low magnetic
field when typical spacing between vortices a 0 essentially
exceeds vortex-vortex ~antivortex! interaction radius j, and
the vortex velocity depends only on the edge screening current that is assumed to be homogeneously distributed across
the sample. The vortex-vortex repulsion, in this case, keep2878
© 1998 The American Physical Society
PRB 58
FLUX-ANTIFLUX INTERFACE IN TYPE-II . . .
ing the number of vortices, cannot play a significant role.
One must take into account both vortex-antivortex annihilation and heat release accompanying this process. We should
also take into consideration heat absorption by the sample
lattice in order to prevent the rise of unlimited temperature.
The vortex-antivortex annihilation obeys the well-known
equations of the recombination theory11,12
]n1
n 1n 2
1div~ n 1 v1 ! 1
50,
]t
t
~1!
]n2
n 1n 2
1div~ n 2 v2 ! 1
50,
]t
t
~2!
t
21
5jv,
v 5mod~ v1 2v2 ! ,
S D
mod~ v6 ! 5 v 6 5 v 6FF exp 2
J5c
the problem becomes one-dimensional, described by the set
of equations
U
.
T
n 1n 2
]n1
]
1
50,
~ n 1 vx1 ! 1
]t
]x
t
~7!
]n2
]
n 1n 2
1
50,
~ n 2 vx2 ! 1
]t
]x
t
~8!
]T
] 2T n 1n 2
5k
1
Q 0 2 g ~ T2T 0 ! ,
]t
]x2
c pt
v 65
~3!
where n 1 and n 2 are the vortex and antivortex densities,
respectively, t 21 is the ratio of recombination for vortices
and antivortices, respectively, j is the cross section of the
annihilation, which is order of the coherence length of the
superconductor, and v6 are the opposite directed vortexantivortex velocities, which in the creep regime are strongly
temperature dependent:
J w 0 exp~ 2U/T !
,
hc
~10!
] n 2 n 1n 2
1
50,
]t
t
~11!
v5vx1 2vx2 .
~12!
Substituting n 2 from Eq. ~11! in the Eq. ~10! and introducing a new variable
where
] Q n 1n 2
5
Q0
]t
c pt
E
~6!
is determined by the energy released by vortex-antivortex
annihilation. Here k is the diffusion coefficient, c p is the heat
capacity, T 0 is the sample temperature in the absence of flux
motion Q 0 is heat released by annihilation of a single vortexantivortex pair, and g 21 5t R is the characteristic time of
temperature relaxation.
The set of Eqs. ~1!–~5! completed by the boundary conditions describes all features of the model.
t
2`
n 1 ~ b ,t 8 !
dt 8 5u ~ b ,t ! ,
t
~13!
one can transform the pair of Eqs. ~9! and ~10! to the following form:
] 2u
] 2 u n 0 ] exp~ 2u !
2
50.
1
v
]t2
]b]t t
]t
~14!
Performing integration over time and solving the obtained
equation by the characteristics method, we obtain for n 1 and
n 2 a shape of the flux-antiflux interface
III. MOVING FLUX-ANTIFLUX INTERFACE
If the magnetic flux enters a superconductor containing
moving vortices of the opposite sign, they interact and annihilate at the interface separating flux-antiflux areas. We show
here that for an extended period, considerably exceeding
some initial transition time of the flux entry, the flux-antiflux
interface travelling wave moves with a constant velocity.
We start from plane flux front dynamics when all of the
functions described by Eqs. ~1!–~5! depend only on a spatial
coordinate parallel to the flux front propagation. In this case
c
u ¹Bu .
4p
n 1n 2
]n1
]
1
50,
~ n 1 v! 1
]t
]b
t
Here, U is a temperature-dependent pinning potential, h is
the viscosity, w 0 is the unit flux, and J is the electric current.
The set of equations @Eqs. ~1! and ~2!# must be completed
by the temperature transfer equation in the form
~5!
J5
~9!
Neglecting in the main order the sample heating by the
vortex-antivortex annihilations, we immediately obtain from
Eq. ~9!: T5T 0 .
In this approximation the pair of nonlinear Eqs. ~7! and
~8! may be solved exactly for the constant vortex-antivortex
velocities. ~This assumption is justified by the solution,
showing that vortex velocity is a constant on the interface
separating flux and antiflux areas.! In the coordinate framework attached to the moving antivortices b 5x2vx2 t, we
obtain, instead of Eqs. ~7!–~9!,
¹3B
, B5 w 0 ~ n 1 2n 2 ! , v6FF 5J3 w 0 / h c. ~4!
4p
]T
dQ
5 k DT1
2 g ~ T2T 0 ! ,
]t
dt
2879
n 1 ~ z ! 5n 0
1
,
11exp~ 2n 0 z / v t !
~15!
n 2 ~ z ! 5n 0
exp~ 2n 0 z / v t !
,
11exp~ 2n 0 z / v t !
~16!
where
z 5x2Vt,
V5
vx1 1vx2
.
2
2880
F. BASS, B. YA. SHAPIRO, I. SHAPIRO, AND M. SHVARTSER
For the oppositely directed flux and antiflux velocities, the
flux-antiflux interface possesses velocity
V5 ~ mod vx1 2mod vx2 ! /2.
The spatial distributions of vortex and antivortex flux densities overlap, forming the interlayer where the vortices of
the opposite signs coexist. The characteristic width of this
region DL may be estimated from Eqs. ~15! and ~16! as
DL. v t /n 0 ;(n 0 j ) 21 , which is a microscopically large area
where the total magnetic induction is suppressed.
B5 f 0 ~ n 2 2n 1 ! 52n 0 f 0 tanh~ n 0 z / v t ! .
v 65
n 20 jw 20
exp~ 2U/T !
4 p cosh2 ~ n 0 z / v t !
h
,
S
D
n 20 jw 20 exp~ 2U/T !
z2
12
.
4p
h
~ DL ! 2
~19!
Namely, at this point our results become asymptotically
exact.
It is interesting to note that even far from the interface
point when u z u @DL, an exact solution of Eq. ~7! n 6 'n 0
practically coincides with the approximate one @see Eqs. ~15!
and ~16!#. This means that in this region the exact solution in
fact does not depend on vortex velocity.
The characteristic size of the region where vortices and
antivortices coexist, DL may be obtained in the general case
from the scaling analysis. Indeed, introducing new variables
in exact equations @see Eqs. ~1!–~4!#
n 1 /n m 5N 1 , n 2 /n m 5N 2 , x/DL5X,
h ~ DL !
,
n 0 w 20
n 2 5n 0 for z .0 and n 2 50 for z ,0
S D
]b
]
]b
5
N
,
]t* ]X
]X
S D
~24!
U U
]N
]
]b
]b
5
b
2 ~ N 2 2b 2 !
,
]t* ]X
]X
]X
~25!
where b is the dimensionless magnetic induction.
These sets of equations have an approximate solution,
b'b 8 z * ,
~26!
in the vicinity of the interface point
Here b 8 is a constant and z * 5X2wt * , where w is the
interface velocity and w< u b 8 u .
In the stationary state one can integrate Eq. ~24!
]b
52I.
]X
N
~27!
Substituting N from Eq. ~27! into Eq. ~25! we immediately
obtain the equation for induction b
S
D
~28!
]b
.
]X
~29!
]
I2
W2
21 bW50,
]X
W2
where
W5b
~20!
where n m is flux density at the interface point, we immediately show that DL is in fact the characteristic spatial parameter of the problem.
It should be noted that the approximating solutions @see
Eqs. ~15! and ~16!# transform into the exact solution of the
problem
n 1 5n 0 for z ,0 and n 1 50 for z .0,
we obtain from Eqs. ~7!–~9!
~23!
Performing the integration we get the differential equation
for b in the form
2
DL5 ~ n m j ! 21 , t/t 0 5t * , t 0 5
b5N 1 2N 2 , N5N 1 1N 2 ,
~18!
which approaches a constant value as B goes to zero at the
interface ( z →0), where
v 6'
sides and their annihilation in the middle point. Introducing
new variables @see Eq. ~20!# in exact equations and supposing that
~17!
The vortex-antivortex velocity in this case has the form
PRB 58
2W1
b
when DL goes to zero.
b
IV. SPATIAL DISTRIBUTION OF FLUX-ANTIFLUX
DENSITIES IN RESTRICTED SAMPLES
The general picture of the flux-antiflux dynamics in the
samples of restricted geometry may be strongly different
from those occurring in the infinite systems. In particular, the
flux-antiflux interface cannot move and must be immobilized
in the stationary state that is formed due to a balance between flux-antiflux entering the sample from the opposite
~30!
This equation may be solved analytically very close to the
interface point where b goes to zero.
Assuming that the vortices and antivortices appear at the
edges of the samples separated by the distance D and assuming the following boundary conditions
~21!
~22!
U U
I
I1W
b3
ln
52 .
2
I2W
3
D
]b
52I, N 2 50 at X52 ,
]X
2
]b
D
52I, N 1 50 at X5 ,
]X
2
~31!
we obtain an asymptotically exact result for magnetic induction at the interface:
b ~ X ! 52I 2/3X for X→0,
~32!
where dimensionless vortex velocity at the interface u 0
5I 2/3 is a constant depending only on vortex/antivortex flow
discharges into the sample.
FLUX-ANTIFLUX INTERFACE IN TYPE-II . . .
PRB 58
FIG. 1. Stationary spatial distribution of the flux-antiflux density
in a superconducting layer.
Returning to the dimension variables, we obtain for interface flux velocity
n 20 jw 20 exp~ 2U/T ! 2/3
I ,
v 6'
4p
h
FIG. 3. Normalized vortex velocity at the flux-antiflux interface
for a stationary exact solution ~curve 1! and for a model with constant velocity ~curve 2!. Both of them are practically constant in this
area.
P~ z !5
~33!
which practically coincides with those obtained for the infinite superconductor @see Eq. ~19!#.
The set of Eqs. ~24! and ~25! with the boundary conditions @Eqs. ~31!# have been solved numerically. The results
are presented in Figs. 1 and 2 where the spatial distribution
of flux-antiflux density and magnetic induction are plotted
for a sample with width D5100DL. The region of vortexantivortex coexistence in this case, DL * '10DL, also qualitatively agrees with those obtained for a moving interface in
the infinite sample. The velocity of the vortices in the interface is shown in Fig. 3. It is clear that a vortex velocity on
the interface is practically constant.
which after some simple transformations reads
G ~ z ,t ! 5
G ~ z 2 z 8 ,t2t 8 ! P ~ z 8 ! d z 8 dt 8 ,
where
E
T ~ z ,t ! 5T 0 1
exp~ i v t1ik z z !
dk z d v ,
i v 2ik z V1 g 1 k k 2z
E
~37!
exp@ i v ~ t2t 8 ! 1ik z ~ z 2 z 8 !#
i v 2ik z V1 g 1 k k 2z
3 P ~ z 8 ! dk z d v d z 8 dt 8 .
Flux-antiflux annihilation, releasing heat, results in a temperature growth in the sample. Supposing that the temperature far from the flux-antiflux interface remains constant, T
5T 0 , and considering the temperature deviation by the perturbation theory, we obtain from Eqs. ~9!, ~15!, and ~16! for
the spatial temperature distribution over the sample
~35!
] G ~ z ,t !
] G ~ z ,t !
] 2 G ~ z ,t !
2V
2k
1 g G ~ z ,t ! 5 d ~ z ! d ~ t ! ,
]t
]z
]z 2
~36!
A. Temperature wave
E
n 1~ z ! n 2~ z !
Q0
c pt
and G( z ,t) is the Green function of the temperature transfer
equation
V. TEMPERATURE DISTRIBUTION
T ~ z ,t ! 5T 0 1
2881
~38!
Evaluating this integral one obtains a spatially inhomogeneous temperature distribution over the sample, depending
only on the travelling coordinate z, in the form
T ~ z ! 5T 0 1
~34!
1
E
2 p exp~ 2 z a 1 !
V A114 kg /V
2 p exp~ 2 z a 2 !
V A114 kg /V 2
2
z
2`
E
`
z
exp~ z 8 a 1 ! P ~ z 8 ! d z 8
exp~ z 8 a 2 ! P ~ z 8 ! d z 8 , ~39!
where
a 1,25
FIG. 2. Magnetic induction across a sample.
V
~ A114 kg /V 2 61 !
2k
~40!
and L 1,2[ a 21
1,2 are the characteristic lengths of the temperature distribution from the interface.
It should be noted that this solution describes the temperature wave, running across the sample with the velocity V.
The shape of the wave depends strongly both on the ratio
L 1,2 /DL and on the interface velocity V.
If the interface width is much smaller than the temperature distribution lengths (L 1,2@DL), one can perform the
integration in Eq. ~39!. The temperature distribution in this
case has the evident form
F. BASS, B. YA. SHAPIRO, I. SHAPIRO, AND M. SHVARTSER
2882
FIG. 4. Spatial distribution of the temperature caused by the
vortex-antivortex annihilation. The shape of the temperature wave
strongly depends on its velocity V changing from symmetrical ~at
low velocity V, A2 kg ! to strongly asymmetrical in the opposite
case ~see curves 1 and 2 in the figure!.
2 p n 20 Q 0
T ~ z ! 5T 0 1
Vc p t 0 A114 p kg /V 2
exp~ z /L 2 !
~41!
T ~ z ! 5T 0 1
Vc p t 0 A114 p kg /V 2
exp~ 2 z /L 1 !
Akg
2k
,
V
L 2.
V
.
2g
T ~ z ! 5T 0 1
exp~ 2n 0 z / v t !
,
g c p t 0 @ 11exp~ 2n 0 z / v t !# 2
E U] ] 88 U
b~ X !
X
DL
Ak / g
~46!
,
where P(X) is defined by Eq. ~35! and DT(X)5 @ T(X)
2T 0 # .
Performing integration in Eq. ~46! we immediately obtain
for DT(X) in the arbitrary units, a result plotted in Fig. 5 that
is qualitatively similar to those obtained for the temperature
wave in the limit of a very slow interface velocity V.
VI. OVERHEATING INSTABILITY
~43!
~44!
In the opposite limit case when the interface width exceeds sufficiently the temperature spatial lengths DL@L 1,2 ,
the integration in the leading order on L 1,2 /DL!1 in Eq.
~39! gives a qualitatively correct result:
4 p n 20 Q 0
Q 0 n 4m j 2 w 20 exp~ 2U/T !
cp
p
h
~42!
to strongly asymmetrical for V@ A2 kg with L 2 @L 1 ,
L 1.
DT ~ X ! 5
l T5
for z .0. Here t 0 [ t (T 0 ).
The spatial temperature distribution is presented in Fig.
4. It is interesting to note that the shape of this distribution
strongly depends on the interface velocity V changing from
symmetrical at low velocities (V! A2 kg )
L 1 5L 2 5
FIG. 5. Spatial distribution of temperature for a stationary fluxantiflux flow strongly depends on the ratio l T 5DL/ Ak / g @here l T
510 ~curve 1!, 3 ~curve 2!; 0.1 ~curve 3!, respectively#.
3exp~ 2 u X2X 8 u /l T ! N 1 ~ X 8 ! N 2 ~ X 8 ! dX 8 ,
for z ,0 and
2 p n 20 Q 0
PRB 58
~45!
simply repeating the shape of the interface.
B. Stationary heating
In real physical systems, flux-antiflux dynamics reaches
its stationary behavior for a relatively short time. Evidently,
this state is provided by an intensive vortex-antivortex annihilation accompanying heat release. This process competes
with heat transport, bringing out the heat from the hot zone
and distributing it across the sample. Using Eq. ~38! in the
static limit, we immediately obtain for the spatial temperature distribution
Let us consider the stability of this vortex-antivortex interface in respect to small deviations from its initial plane
shape. In the framework attached to the antivortices, one can
look for the solutions of Eqs. ~1!, ~2!, and ~5! in a stationary
state in the form
n 1 ~ z ,y,t ! 5n 1 ~ z ! 1C ~ z ,y,t ! ,
n 2 ~ z ,y,t ! 5n 2 ~ z ! 1U ~ z ,y,t ! ,
S
mod v' v 0 11
D
S
~47!
D
D ~ z ,y,t !
D ~ z ,y,t !
, t 21 ' t 21
11
,
0
T0
T0
v 0 5mod v~ T 0 ! ,
T5T 0 ~ z ! 1
S
~48!
T0
D ~ z ,y,t ! ,
U
vy ' v 1 11
D
D ~ z ,y,t !
,
T0
~49!
~50!
where we assume that vortex velocity v1 5(vx1 ,vy1 ) also
possesses the component parallel to the interface vy and
vy1 5 v 1 ! v 0 . For simplicity we neglect the vy2 component
for antivortex motion.
Here we also neglect the influence of temperature fluctuations on heat capacity c p and relaxation coefficient g because
PRB 58
FLUX-ANTIFLUX INTERFACE IN TYPE-II . . .
their calculations do not result in essential effects. We also
neglect in the main order the change of the average temperature in the flux front area.
Substituting these relations in the initial set of Eqs. ~1!,
~2!, and ~5!, we obtain after linearization the set of equations
for deviations:
S
D
]D
] 2D ] 2D
5k
1
2gD
]t
] y 2 ]z 2
~51!
UQ 0 @ n 1 ~ z ! U1n 2 ~ z ! C1n 1 ~ z ! n 2 ~ z !~ D/T 0 !#
1
,
T 0t 0c v
~52!
S
D
]C
] C n 1~ z !
]D
]D
]C
1v0
1
1v1
1v1
v0
]t
]z
T0
]z
]y
]y
1
@ n 1 ~ z ! U1n 2 ~ z ! C2n 1 ~ z ! n 2 ~ z !~ D/T 0 !#
t0
k c pT 0
!1.
v 0 j Q 0 Un m
~53!
V 3 1V 2 @~ iq11 ! 1Aq 2 1 g * 2 P * #
1V
S
iq
P*
1 ~ Aq 2 2 P * 1 g * !~ iq11 ! 1iq P * 1
2
2
S
D
SD
C ~ z ,y,t !
C
U ~ z ,y,t ! 5exp~ v t1ik y y2 a z 2 /2! U ,
D ~ z ,y,t !
D
S
n mQ 0U
~ U1C ! 50,
t 0c vT 0
D
nm
nm
nm
v 1ik y v 1 1
C1
U1
ik v D50,
t0
t0
T0 y 1
S
v1
D
n 2m
nm
nm
U1C
1
D50,
t0
t 0 t 0T 0
~61!
where
V5
vt0
k yv 1t 0
2knm
2t0
, q5
, A5 2 , P * 5 P
,
2n m
2n m
nm
v 1t 0
g *5 g
2t0
.
nm
~62!
Representing V in the form
V5V 1 1iV 2 ,
~63!
we immediately obtain for the real and imagined parts of the
V
V 31 23V 1 V 22 1 ~ V 21 2V 22 !~ Aq 2 2 P * 1 g * 11 ! 22V 1 V 2 q
~55!
we obtain from Eqs. ~51! and ~54! a set of equations depending only on z.
Multiplying all of the equations by exp(2az2/2) we obtain after the integration on z:
~ v 1 g 1 k k 2y 2 P ! D2
~56!
~57!
~58!
1V 1 ~ Aq 2 2 P * /21 g * ! 2V 2 q
S
P5
t 0 T 20 c p
.
~59!
Here, n m is a vortex density in the interface point.
It should be noted that these equations have been obtained
for variational parameter a @1 because all of the integrals
containing the trial function Eq. ~55! rarely depend on a for
a >1 and drop to zero for the a ,1. We can also neglect in
the low-temperature region the term k ] 2 D/ ]z 2 in comparison with heat source term
D
1
1 g * 1Aq 2 50,
2
~64!
3V 21 V 2 2V 32 1 ~ V 21 2V 22 ! q12V 1 V 2 ~ Aq 2 2 P * 1 g * 11 !
1V 1 q
1
S
D
1
1 g * 1Aq 2 1V 2 ~ Aq 2 2 P * /21 g * !
2
q
~ g * 1Aq 2 ! 50.
2
~65!
These equations may be solved exactly for extremely long
waves (q→0). Looking for the real and imagined parts of V
in the form
V 1 5aq 2 1cq 4 , V 2 5dq1 f q 3 ,
where
Q 0 Un 2m
D
iq
~ Aq 2 1 g * ! 50,
2
] U @ n 1 ~ z ! U1n 2 ~ z ! C1n 1 ~ z ! n 2 ~ z !~ D/T 0 !#
1
50.
]t
t0
~54!
It is evident that all of the perturbations of the interfacelike
solution far from the flux-antiflux interface cannot result in
front instability. Therefore, one can assume that these perturbations must be localized on the boundary.
Looking for the solution in the form
~60!
This inequality satisfies for t D @t A , where t D and t A are
the characteristic times of diffusion and annihilation processes, respectively.
The conditions of self-consistency for Eqs. ~56!–~58! result in the following equation for the increment in the dimensionless variables:
1
50,
2883
~66!
we obtain for V 1 in the limit of small g * !1/4A!1(q
→0) ~t A /t R !1 and Ak /t A @ v 1 @ Ak /t R !:
V 15
S
D
1 P*
1
2
q 4.
2 21 q 2
4 2g*
16g * 2
~67!
The shape of the increment curve V 1 (q) is presented in Fig.
6. Corresponding with the general theory, the instability develops for positive V 1 . It immediately results in the threshold value of the parameter P * for instability appearance:
P * .2 g * 2 .
F. BASS, B. YA. SHAPIRO, I. SHAPIRO, AND M. SHVARTSER
2884
PRB 58
much faster then the velocity of the vortex-antivortex boundary. The second one is the case where the interface width
sufficiently exceeds the temperature spatial lengths DL
@L 1,2 . In this case the temperature relaxation processes are
very rapid and the heated region is confined inside the interface area.
Temperature distribution becomes strongly asymmetrical
for essentially large front velocities V@ A2 kg . In this case
the temperature tail behind the interface stretching on the
distance L 2 sufficiently exceeds temperature zone L 1 in front
of the interface, where
L 1.
FIG. 6. The rate growth of instability Re V vs wave vector q. If
the flux velocity exceeds some critical value, the increment of the
flux-antiflux interface perturbations manifests its instability.
In the dimensional variables @see Eq. ~62!# this condition
reads
v 0> v c5
A
c p T 20 g 2
Q 0 Un 3m j 2
.
~68!
The wave vector q m at which V 1 vanishes sets a spatial scale
for the problem. For velocities of the flux exceeding this
critical value v c there is a band of unstable wave vectors
where increment of instability is positive, and a critical
length scale defined by the wave vector q c of the fastest
growing mode
q 2c 52 g * 2
S
D
P*
21 .
2 g *2
l c 52 p /k c .
Av 2 / v 2c 21
@a 0 .
~69!
~70!
This is exactly the condition that we need in order to
justify using the hydrodynamics approach.
VII. CONCLUSION
Thus, dynamics of the vortex-antivortex flux in the type-II
superconductor possesses the following most important
properties:
~1! For sufficiently small velocity of the vortices v , v c
flux-antiflux interface propagates with velocity V5(mod v1
2mod v2 )/2, where v1,2 are the flux/antiflux velocities.
~2! The characteristic size of the region where vortices
and antivortices coexist DL may be macroscopically large
DL;a 20 / j @a 0 , where a 0 and j are the intervortex distance
and linear cross sections of the vortex-antivortex annihilation, respectively.
~3! The main features obtained for the model of constant
vortex velocity remain correct for exact equations due to a
scaling character of the problem.
~4! The spatial temperature distribution ~Fig. 4! becomes
symmetric in two limits. The first one is the limit of low
velocities V! A2 kg , when the heat spreading velocity is
L 2.
V
.
2g
~71!
~5! If the vortex flux velocity on the interface exceeds its
critical value v 0 . v c , the flux-antiflux boundary manifests
the instability. The instability is caused by vortex-antivortex
overheating exceeding thermal absorption in the sample.
This type of instability is well known in plasma physics as
‘‘overheating instability’’.13,14 The critical velocity of this
instability, determined by Eq. ~68! may be obtained simply
in the following manner. The temperature increasing during
the time of unit annihilation act t A , may be estimated as
Q 0 n 0 /c p t A @see Eq. ~7!#, which must be multiplied by the
number d N of the unit acts during the heat relaxation time t R
where d N;t R /t A . The relaxation caused by the sample lattice is the competing process in the system. The temperature
relaxation for time t R may be estimated as T 0 /t R . It is evident that the overheating instability arises under the condition
In the dimensional variables we get
v 1t R
2k
,
V
DQ t R Q 0 n 0 T 0
[
2 .0,
tR
t A c pt A t R
~72!
t R 5 g 21 , t A 5 ~ n 0 v 0 j ! 21 ,
~73!
which is equivalent absolutely to those defined by Eq. ~68!.
~6! The most rapidly growing mode of instability occurs
at wavelength l c . It seems reasonable to presume that patterns emerging from this instability will, at least initially,
have a characteristic size of order l c @a 0 in which case l c
may be macroscopically large.
~7! The increment of corrugation instability V 1 (q) shows
that in the domain of instability the growing modes develop
in a wide region of the wave vectors. The wavelength l m at
which V 1 vanishes, sets a length scale for the problem. The
set of Eqs. ~67!–~69! tells us that l c 5&l m , and the minimal wavelength of the instability has the same order as the
rapidly growing mode l c . The existence of this length has a
very clear physical explanation. It appears as a balance between the excess heat c p DQl m released in the region of the
size of the order of l m for the characteristic time t R and the
heat flow s y carried by the vortices and directed along the
interface, and bringing this heat out from this hot domain due
to the temperature gradient in this direction. The balance
relation gives the condition of instability
DQl m . ~ v 1 t R ! 2 T 0 /l m ,
where
~74!
PRB 58
FLUX-ANTIFLUX INTERFACE IN TYPE-II . . .
2885
s y 52c p ~ v 21 t R ! ¹ y T
c p 5N ~ e F ! D 0 ~ D 0 /T ! 3/2exp~ 2D 0 /T ! , g 5 g e2ph 5T 3 / v 2D .
and v 21 t R is the heat diffusion coefficient of the vortices,
sliding along the interface.
Substituting DQ from Eq. ~72! in Eq. ~74! we immediately obtain for l m the result of the linear theory of instabilities defined by Eq. ~69!.
~8! Instability appears only in a respectively narrow temperature ‘‘window’’ determined by inequality v 0 (T 0 )
. v c (T 0 ), resulting in the following relation @see Eqs. ~66!–
~68!#:
Here v FF ; w 0 J c /c h , J c ;c w 0 n 2m j , D 0 and v D are the superconducting gap and Debye frequency, respectively, J c is
the critical current, and N( e F ) is the density of states at the
Fermi level.
v 2FF Q 0 n 3m j 2
T0
ln
.2,
U
T 0c p~ T 0 ! g ~ T 0 !
~75!
where
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ACKNOWLEDGMENTS
We would like to thank V. Vinokur and M. Bazilevitch
for helpful discussions. This work was supported by the
Israel Ministry of Sciences and Arts, the German-Israel
Foundation and the Israel Academy of Sciences. We would
also like to thank the Bar-Ilan Minerva Center for Superconductivity for permanent support.
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