Simulations of magnetization process in superconducting films

SIMULATIONS OF ~ IAGN,E'~IZAItON IN SUPgR~ONDUCTING FILMS Ry~e KA~

KOCE~5

Yc~hihL~ ENOMOTO ~}, ~ d S,ada~mehi MAERAWA ~

We study the d y ~ a m i ~ of flux stalin in t y ~ l l 8t~perce~d~eto~':4, ~'v~s*-do~ the time 4ep<~dsr~t Gmzeb,~g Landat~ {TDGL) ~ u a t i e n inct,ding the thermal ~uctuatk,~s C o m p ~ c : sim~.~ati.on~ f~,.,r ~.~.~',~t~ < ~ e~e perfotm~t to d i s e ~ the time--dependent behavior of supercoadt~cti~g ~t ~te~ wit h am~ w d h
T h e d~seovety of the high-~empeeature oxide suo in the normM s ~ t e

rv~p.c
stay

[)

the m~m;~i

them are teh to be examined in detail theoreticMb'

staa~

and

and expenmentaRy

r,:h~tion,

pe~onduc~om t has intensified i n t e n t s t a t < ~ ~,

in the vortex

Various new vortex stat,~ m the oxide suo

perco~tduc~o~s were s u ~ t ~ x | 3 " ~

However, some of

The ~ime~dependen~ Ginzburg-

,~md # a ~

cond~cii~t}',

s~atc dffTusi
re~pc<~',d) ,

and

ha~, ~ thr.

l{~e

( ~s the coherertc~

, /i7 , t ~ f r' t"~ ..... ~,:~J ( ............... !

t,:!>(r ......r'i~it ......t'i

~

Landa~t (TDGL) equation s ~ one of the mint ~.tsefu[ txluat~ons [o~ s ~ d y i n g the dFnamics of super¢onduc~ tots.

Here, we p r o p ~ e a new simulation method for

studying the dynamics in a m ~ n e t i e field. The thermal


'

flucluations ~ are expiieit~y included in the equation

. rr

.

Solving the TDGL equation numerically, we s t u d y the time evolution of the superconducting state. T h e nucleation and annihilation of vo~ices and their motion are examined with and without art external magnetic

>gu{,:~ - T}

T~e TDGL eq~.~a~ion L~ soIv<~t

together with the M&,
fieM

Let us introduce a garage t~a~sforma~on of the v,~<-

The TDGL equation is a partial differential equation for the space and time dependence of the c o m p b x order parameter A and is conveniendy written in the normalized form adapted by Hu and Thompson r, as

tot and s c a b . potentials by r-placing A with A - Vk and ~, with ~, + by ~he p h ~ e

_:~a_, i---~-

An-

~

flea

A

~n ¢~x~erna[ c u r t < h i

tR o~-c~r" i ~ ;0

- ,~-= (lal = - ~) ~ + f(~,el,

te~aie

~.:a,:a~-~ t h e . - C a l f , :
,.-a=~z~.i.;i~s ~

-,~ ~:~at w~-..:,. ~:<:;I<~1 ~i

:4 .a C,k

-

...

i t !~: : : p , c - < > : . . > t

c~e~,i<>ws, i,:i:~',h

ir~

~: !:~ (

t = ~1~. ~ime in units of t<;L/12, an:i the v<,c~,>~ p ~.ential in u n i t s

c 8t

--

~exp without

-

respectively, accompani~]

~edefini~ion of order parameter, A

....; , / ~ H . ~ I

>,~>~ct!ve!y

v~here ~ ~:~

positive constant, which is after~aEd defined ~ ~* make t h e numerical calculation e~cient. H ~ m ~: ~: th~

a_ Re A* '

0921-4534/91/$03.50

i

A bc

A

b~reA-' ~ "

@ 1991 - Elsevier Science Publishers B.V.

GinzBu~g-Landau pa::<:':~ t, :..o. =.~, 'f

All fi~hts ~.served.

1718

P~ Kato et al. / Simulations of magnetization process

We study a thin fihn of the type-II superconductor with the xy plane and a static magnetic field applied perpendicular to the xy plane (z-direction). The thickness of the film, d, is in the range d ~ ,k so that the TDGL equation is independent of z-coordinate. The periodic and free boundary conditions are introduced in the x and y directions, respectively. Therefore, at the film edge in the y direction,we have = 0, hc

n

for the order parameter where the suffix n denotes the normal direction at the boundary, and

FIGU RE 1 The spatial pattern of the magnetic flux density in h~-'0.25. In this figure, the maximum value is 0.25 and the minimum value is 0.

'V x A = H e ,

for the vector potential where He is the external magnetic field, which is applied along the z-direction, ~,

In Fig. 1. we show the spatial pattern of the magnetic flux density in h~=0.25 obtained in the simulation

and is given by He=he(x, y) ~. For the numerical calculation of the TDGL equation, the film is divided into a square lattice with N 2

superconducting properties. The magnetic properties

lattice points, N being 60. By setting a to be 2, the size

in inhomogeneous samples and the interaction between

of the film is 30~x30~ in the physical units. We have

electric current and vortices are now in progress.

used several sizes of the unit cell and found that this size of unit cell gives the convergent results in the cases studied in this paper. The order parameter is divided into real and imaginary parts. The TDGL equation is numerically integrated with the time step At=0.05. We have fixed ~ to be 2. The upper and lower critical lnagnetic field, He2 and Hat, are therefore 1 and 0.083, respectively, in our units. Leaving the details of the simulation technique in Ref. 8, we summarize the simulation results: 1. Throughout the simulation each vortex has a quantized magnetic flux ~o as is expected. 2. The nucleation process of' ae superconducting state at he=O is accompanied by the nucleation and annihilation of vortices. 3. In a low magnetic field (He1
method. This method is applied to the study of many

References 1. J. G. Bednorz and K. A. Miiller, Z. Phys.

B64

(1986) 189. 2. See, for example, A. P. Malozemoff, Physical Prop-

eriies of High Temperature Superconduciors, ed. D. M. Ginzburg (World Scientific Pub., 1989) vol.1, pp. 971. 3. D. S. Fisher, P. A. Fisher, and D. A. Huse, Phys. Rev. B43 (1991) 130. 4. D. R. Nelson, Phys. Rev. Left. 60 (1988) 1973. 5. A. Schmid, Phys. Rev. 180 (1969) 527. 6. M. Tinkham, Introduction to Supercond,,ctivity (McGraw-Hill, New York, 1975). 7. C.R.Hu and R. S. Thompson, Phys.

Ray.

B6

(1972) 110.

8. R.Ka, to, Y. Enomoto, and S. Maekawa: to be published.