On the magnetic flux — branching instability in superconductors

North-HolClandPhysi 235-240 ca (1994)3017-3018

ON

THE

I.L.

MAGNETIC

FLUX

PHYSlCA (~

-

BRANCHING

INSTABILITY

IN

SUPERCONDUCTORS 0

Maksimov

Department of Theoretical Physics, Nizhny Novgorod University 23 Gagarin ave., Nizhny Novgorod, 603091, Russia. Novel instability representing an avalanch-like penetration of the magnetic flux into superconductor is explained in terms of thermomagnetic shock wave propagation. It is shown that the long-wavelength instability of the critical state with respect to the two-dimensional thermal and electromagnetical disturbances may be responsible for the pattern formation in the course of the magnetic flux penetration.

1.

the phonons ensemble [3]. This leads to the energy flux relaxation for the time interval T determined by the rate of nonconserving momentum collisions (due to e.g. umklapp processes) and results in the nonstationary modification of the Fourier law for the heat flux Q [4]: dQ -7 + Q = - K (VT)

INTRODUCTION

The theoretical model is proposed to describe the magnetic flux branching instability, discovered recently [i] Such an instability manifests itself as an extremely fast penetration of the magnetic flux into type-If superconductor in the mixed state. The explanation is formulated in the framework of the thermomagnetic shock waves concept proposed for the first time by the author in [2]. Such excitations may occur in superconductors due to thermal softening of the pinning force and represent a kinklike profile of the magnetic field, propagating with the constant speed inside the sample. 2. O N E - D I M E N S I O N A L

(here

K is the heat conductivity) Consider plane semiinfinite sample (x>O), placed in the magnetic field. Searching the solution of the Maxwell and the modified heat diffusion equations combined with the critical state equation j(T,E) = j ( T ) c

Research

N

93075

N

SHOCK WAVE

supported

by

93-02-16876), and

by

the

the by

International

Council

o

the superconductor) in the running wave form H = H(z) (where z=x-vt) we find for the superconductors with Research for

Science

0921-4534/94/$07.00 © 19'94 - Elsevier Science B.V. All rights reserved. SSDI 0921-4534(94)02074-4

E ef

conductivity in the flux-flow regime T is the equilibrium temperature of

Fundamental the

o

( a - is the thermal softening coefficient of the magnetic flux pinning force, ~efis the effective

The nontrivial feature of the phenomenon of superfast magnetic flux penetration is the partial thermodynamic equilibrium reached in

(grant

- a(T - T ) + ~

o

HTSC

Foundation o£

Foundation.

the

Russia

of

the under

Russla Project

3018

I.L. Maksimov/Physica C 235 240 (1994) 3017 301,'¢

high m a g n e t i c d i f f u s i o n coefficient D (compared with the thermal diffum

sion one

D

): ~ = D t

/ D t

<< 1

the

m

d e p e n d e n c e of the wave p r o p a g a t i o n velocity v on the b a c k g r o u n d field value H inside the S h u b n i k o v phase: 0

a v

Dm

-

H o

2 c p

(here p is the heat capacity c o e f f i c i e n t of the material). The m a g n e t i c field d i s t r i b u t i o n in the s w i t c h - w a v e is g i v e n by the relation z = A {[i + 2 ( v / V ) 2 ] i n ( l - h ) where

k = D m

and

V=(D

/v 1/2

;

h =

- In(h)}

H/H

< 1

;

o

/T) t sound velocity of condensate. Numerical

is

the

second

6E(x,y,t)

the m a t e r i a l s

with D ~i0+

= 6E

exp[At

+i(qx+ky)/l]

o

For latter s u b s t i t u t i o n it is easy to find the dependence of the disturbances' increment k on the value of the w a v e n u m b e r k of the longitudinal modulation of the critical state: _

Dm 2 1

( ~

q2 _ k2 )

_

Taking into account the values of and q near the s t a b i l i t y t h r e s h o l d for the one-dimensional critical state [5]:

superconducting estimate gives

the velocity v a l u e v ~ lO s + 1 0 6 c m / s comparable with the value observed experimentally. The width of the onedimensional front is of the order k~

lO-4cm for

equation is valid on the initial stage of the magnetic flux motion, which is d e s c r i b e d by disturbances 6E(x,y,t) of type:

~ ~ = ~2/4

;

m

2

q

2 = ~ /(1 ~ qmax m

+ ~

)2

one finds that the long-wavenumber m o d u l a t i o n instability, similar to that, actual for the solidification problem [6], may occur p r o v i d e d

m

k2

I00 cm2/s

< k2 ~

2~

o

3.

TWO-DIMENSIONAL INSTABILITY THE CRITICAL STATE

OF

The scale modulation

of Ay:

Ay = i/ ~ To d e s c r i b e the flux branching phenomenon we shall analyse the s t a b i l i t y of the critical state with respect to small two-dimensional disturbances. By l i n e a r i z i n g system evolution e q u a t i o n s and e x c l u d i n g variable T we come to the e l e c t r o m a g n e tic d i f f u s i o n e q u a t i o n with activity: 3E =

D

[ ( ~/I 2) 3E

+ V 2 3E

]

m

here

~ = 4~j

(T)al c

~(c ~) - is

a

o

parameter, d e s c r i b i n g the stability c o n d i t i o n for the critical state [5] and 1 = A - is the space scale of the critical state localisation in the t r a n s v e r s e (with respect to the M e i s s n e r currents) direction. This

the -I/4

longitudinal

~ i

being appreciably larger than the shock-wave front width can be of the order quite

Ay ~ 10-3+ lO-4cm, w h i c h close to the data [i].

is

REFERENCES i. P.Leiderer et. al. Phys. Rev. Lett., 71 (1993) 2642. 2. I.L.Maksimov, J.Phys. D., 21 (1988), 251. 3. R.J.Hardy, Phys. Rev. B2 (1970)1193 4. D.D. Joseph and L. Preziosi, Rev. Mod. Phys. 61 (1989) 41. 5. R.G. Mints and A.L. Rakhmanov, Rev. Mod. Phys. 53 (1981] 551. 6. J.S.Langer Rev. Mod. Phys.52(1980)l.