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THE GENERATION OF SOLUTIONS TO THE
GINZBURG-LANDAU EQUATIONS FOR A
DISTORTED VORTEX LATTICE
J. Evetts
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C6, suppliment au no
8, Tome 39, aozit 1978, page C6-621
T H E GENERATION O F SOLUTIONS T O T H E GINZBURG-LANDAU EQUATIONS FOR A DISTORTED
VORTEX LATTICE
J.E. E v e t t s
Dept.
ofhetaZZurgy and MateriaZs Science,
Pemb~okeSt., Cambridge,
U.K.
Rdsumd.- Nous d6montrons que l a s o l u t i o n d e s 6quations d e Ginzburg-Landau pour un r6seau d i s t o r d u de l i g n e s de f l u x d r o i t e s e t p a r a l l s l e s peut d t r e obtenu simplement en u t i l i s a n t d e s f o n c t i o n s e x p l i c i t e s q u i i n s s r e n t ou e n l s v e n t d e s l i g n e s de f l u x du rdseau. L ' e x p r e s s i o n du paramstre d ' o r d r e e s t l a msme que c e l l e d & r i v & e par Brandt.Nous d6montrons que l e s hypothsses e t approximations d e l a d 6 r i v a t i o n f a i t e p a r Brandt dqui- v a l e n t 1 l l h y p o t h S s e d e l a v a l i d i t 6 de l a premiSre r e l a t i o n d'Abrikosov dans l a premisre
6quation de Ginzburg-Landau. La v a r i a t i o n d e t a i l l e e de l a p e r t u r b a t i o n du champ magn6ti- que ne s ' o b t i e n t pas d'une manisre e x p l i c i t e . Mais nous dgmontrons q u ' l longue p o r t 6 e e l l e e s t d 6 c r i t e p a r 1 ' 6 q u a t i o n d e London.
A b s t r a c t
.-
I t i s demonstrated t h a t t h e s o l u t i o n of t h e Ginzburg-Landau e q u a t i o n s f o r a d i s t o r d e d l a t t i c e of s t r a i g h t p a r a l l e l v o r t i c e s can b e generated i n a simple way by t h e u s e of e x p l i c i t f u n c t i o n s t h a t i n s e r t o r remove v o r t i c e s from t h e l a t t i c e . The expres- s i o n f o r t h e o r d e r parameter i s t h e same a s t h a t derived by Brandt. The assumptions and approximations made by Brandt i n h i s d e r i v a t i o n a r e shown t o be e q u i v a l e n t t o assuming t h e v a l i d i t y of t h e f i r s t Abrikosov r e l a t i o n i n t h e f i r s t Ginzburg-Landau e q u a t i o n . The d e t a i l e d v a r i a t i o n of t h e magnetic f i e l d p e r t u r b a t i o n i s n o t obtained e x p l i c i t y , however i t i s shown t h a t a t long ranges i t i s d e s c r i b e d by t h e London e q u a t i o n .INTRODUCTION.- It i s w e l l known t h a t t h e s o l u t i o n terms i n t h e GL e q u a t i o n s a r e t h e r e f o r e e s s e n t i a l of t h e Ginzburg-Landau (GL) e q u a t i o n s f o r any s t r i c -
t l y p e r i o d i c a r r a y of v o r t i c e s can b e obtained v e r y a c c u r a t e l y n e a r t h e upper c r i t i c a l f i e l d , BcZ, by s o l v i n g t h e l i n e a r i z e d e q u a t i o n s / I - 3 / . Recently Brandt has shown t h a t t h e l i n e a r i z e d e q u a t i o n s g i v e a t o t a l l y inadequate d e s c r i p t i o n of t h e d i s t o r d e d v o r t e x l a t t i c e / 4 , 5 / . T h i s f a i l u r e has a r a t h e r s i m - p l e o r i g i n . When t h e l i n e a r i z e d e q u a t i o n s a r e used t o c a l c u l a t e t h e p e r t u r b a t i o n of t h e o r d e r parameter a r i s i n g from a l o c a l d i s t o r t i o n of t h e l a t t i c e i t i s found t h a t t h e p e r t u r b a t i o n i s long r a n g e , ( % l / r ) . T h i s r e s u l t s i n a divergence of t h e energy f o r any- non-periodic d i s t o r t i o n . Even f o r t h e case of an i n f i n i t e s i m a l displacement of a s i n g l e v o r t e x t h e
energy is found t o d i v e r g e , ( t h i s i s i m p l i c i t i n t h e r e s u l t s of Kogan
1 6 1 ) .
I f t h e non-linear terms a r e i n c o r p o r a t e d t h e long range p a r t of the o r d e r parameter p e r t u r b a t i o n
i s e f f e c t i v e l y "screened" and t h e d i s t o r t i o n energy becomes f i n i t e . The d i s t u r b a n c e caused by a l o c a l d i s t o r t i o n now f a l l s away e x p o n e n t i a l l y o u t s i d e a c h a r a c t e r i s t i c s c r e e n i n g l e n g t h L A corresponding
JI'
a n a l y s i s of t h e d i s t u r b a n c e of the magnetic f i e l d u s i n g t h e second GL e q u a t i o n shows t h a t t h e magnetic f i e l d p e r t u r b a t i o n a l s o h a s a c h a r a c t e r i s t i c range
Lh
which i s g r e a t e r than L ( i n t h e r a t i of i ~
JI'
where K i s t h e GL parameter). The non-linear
i n any q u a n t i t a t i v e treatment of n o n - p e r i o d h vor- t e x l a t t i c e d i s t o r t i o n and l e a d t o r e s u l t s t h a t a r e of c e n t r a l importance f o r t h e understanding of a number of d i f f e r e n t problems r e l a t i n g t o t h e f l u x v o r t e x l a t t i c e . There a r e s t i l l d i f f i c u l t q u e s t i o n s t h a t r e l a t e t o t h e e l a s t i c c o n s t a n t s of t h e l a t t i c e and i t s response t o pinning c e n t r e s of v a r i o u s r a n g e s / 5 , 7 , 8 , 9 / . Also t h e r e a r e d i f f i c u l - t i e s i n t r e a t i n g q u a n t i t a t i v e l y f l u x l a t t i c e de- f e c t s and d i s l o c a t i o n s and t h e i r r e l a t l o n t o vor- t e x c u t t i n g a t c r o s s o v e r s /7/.
The approach used by Brandt depends on ex- p r e s s i n g t h e v o r t e x displacement f i e l d a s a sum of F o u r i e r components and c a l c u l a t i n g a c o r r e c t i o n t o t h e l i n e a r i z e d r e s u l t s f o r each p e r i o d i c component. The c o r r e c t i o n i s t h e n expanded i n powers of t h e amplitude of the displacement and c a l c u l a t e d t o f i r s t o r d e r i n the displacement. To achieve a so- l u t i o n a number of averages and approximations a r e made; a l s o t h e most g e n e r a l r e s u l t involves a t l e a s t one i n t u i t i v e s t e p . The treatment given here i s not completely r i g o r o u s , t h e main aim has been t o emphasise t h e p h y s i c a l e f f e c t of t h e non-linear terms. The main r e s u l t of Brandt's treatment i s
reproduced d i r e c t l y and t h e approximations inherent i n i t s d e r i v a t i o n a r e expressed i n an a l t e r n a t i v e
way. one o b t a i n s
v~~
= K2w0(e2g-~) THE G-L EQUATIONS.-If t h e v o r t i c e s a r e s t r a i g h ~and p a r a l l e l , t h e magnetic f i e l d = (O,O,hZ) ,and t h e G-L e q u a t i o n s c a n b e e x p r e s s e d i n t h e f o l l o - wing form ( i n d i m e n s i o n l e s s u n i t s ) ,
~ 2 ( 1 / 2 i n w) + K~ = ~~w + [ K ~ ( v ~ ) ~ - ( ~ w ) ~ / ~ + ] / w ~ (1)
-
v2h+
wh = -(Vh) (Vllnw) (2) Here ,o=f2=lY12 t h e s q u a r e of t h e modulus of t h e o r d e r parameter measured i n u n i t s o f Y f o r h=O.The system of u n i t s corresponds t o measuring l e n g t h s i n u n i t s of h,
where h i s t h e p e n e t r a t i o n d e p t h ; and t h e magnetic f i e l d i n u n i t s ofK H ~ ,
where Hci s t h e thermodynamic c r i t i c a l f i e l d .
I f t h e e q u a t i o n s a r e l i n e a r i z e d w i t h r e s - p e c t t o f , t h e second e q u a t i o n y i e l d s t h e f i r s t Abrikosov i d e n t i t y , / 1 , 2 , 5 , 6 / ,
h = H - w/2K (3)
where i s a c o n s t a n t . The f i r s t e q u a t i o n t h e n beco- me s
~ ~ ( ~ / ~ ~ n w )
+
K~ = 0 (4)SOLUTIONS OF THE LINEARIZED EQUATIONS.- The forma- l i s m f o r g e n e r a t i n g t h e s o l u t i o n f o r an a r b i t r a r i l y d i s t o r d e d v o r t e x l a t t i c e i s p a r t i c u l a r l y s i m p l e i n
t h e c a s e of t h e l i n e a r i z e d e q u a t i o n s . The o r d e r pa- r a m e t e r f o r t h e d i s t o r t e d l a t t i c e i s e x p r e s s e d i n t h e form w=woe2gwhere w o i s t h e s o l u t i o n f o r t h e u n d i s t o r d e d l a t t i c e and g i s an undetermined func- t i o n . S u b s t i t u t i o n i n (4) t h e n g i v e s
Q wo)
+
vZg
+ K~ = 0 Whence v2g = 0 (5) The u n d i s t o r t e d s o l u t i o n can t h u s be m u l t i p l i e d by any eZg(where = 0) t o g i v e a new s o l u t i o n . The s o l u t i o n of (5) f o r a l i n e charge d e n s i t y , P , a tr
h a s t h e e f f e c t of i n s e r t i n g ( p = - I ) , o r removing( p = + l ) a f l u x v o r t e x a t t h e s i t e of t h e l i n e charge. Denoting t h e s e s o l u t i o n s by g + ( r ) and g-(5)respec- t i v e l y , w f o r an a r b i t r a r y d i s t o r d e d l a t t i c e beco- mes w =
w a g
e x p ( Z g - ( ~ ~ ) )If
e x p ( 2 g + ( ~ ' ~ ) ) (6) where t h e s e t of v o r t i c e s a tj
a r e d i s p l a c e d t o r'". For t h e d i s p l a c e m e n t of a s i n g l e v o r t e x from t h e o r i g i n t o s = ( s , O ) , -s ij = wo(lr-11
l r I ) 2 = w O ( ~ - ~ x s / r 2 ) + ~ ( s 2 )lrl
>> (7) SOLUTIONS OF THE NON-LINEAR EQUATIONS.- The f i r s t GL e q u a t i o n y i e l d s B r a n d t ' s r e s u l t d i r e c t l y i f h ( i n e q u a t i o n ( 1 ) ) i s assumed t o s a t i s f y ( 3 ) . Then v 2 ( l I 2 inw)+lc2=K2w, s u b s t i t u t i n g a s b e f o r e w=woe2g T h i s e x a c t e q u a t i o n t a k e s a p h y s i c a l l y s i m p l e form f o r s m a l l g g i v i n gv~~
= 2 ~ ~ T h i s w ~i s ~t h e Tho- . mas-Fermi s c r e e n e d p o t e n t i a l ( w i t h a non-udiform s c r e e n i n g c o n s t a n t ) . The a v e r a g e s c r e e n i n g cons- t a n t 2 ~ 2 < w ~ > i s a good approximation g i v i n g g- =+_
kO(kyy) where K O i s a Hankel f u n c t i o n and + k2=1I2 = ~ K ~ < W ~ > . Equation (6) s t i l l a p p l i e s u s i n gY L
t h e new s o l u t i o n s f o r g- ( r ) . I f a s i n g l e v o r t e x
i s d i s p l a c e d a s b e f o r e :o s = ( s , O ) , we have
Equation (11) i s e q u i v a l e n t t o t h e c o r r e c t i o n d e r i - ved by Brandt / 5 / e q u a t i o n s (46,47). Brandt i n t u i - v e l y g e n e r a l i z e s h i s e x p r e s s i o n t o t h e form (91,
( s e e 1 5 1 , e q u a t i o n (55). The d e r i v a t i o n g i v e n h e r e c o n f i r m s t h i s s t e p and c l a r i f i e s t h e assumptions i n h e r e n t i n t h e approximations made by Brandt.
The magnetic f i e l d c a n n o t b e determined u s i n g ( 3 ) , i n s t e a d ( 2 ) must b e s o l v e d . The form of
(2) i s c l o s e l y a n a l g o u s t o t h e London-Abrikosov e q u a t i o n / 1 0 , 2 , 4 / . I n e q u a t i o n (2) we may r e g a r d t h e term -(Vh) (Vllnw) a s a summation of a p p r o p r i a - t e l y smeared o u t d e l t a f u n c t i o n s over a l l t h e vor- t i c e s . I f t h e f i e l d i s e x p r e s s e d a s h=ho+h where
g
h i s t h e u n p e r t u r b e d f i e l d we o b t a i n t h e simple London form -V2h +wh =O o u t s i d e t h e r a n g e Ly. Ave-
g g
r a g i n g as b e f o r e w e s e e t h a t h has a c h a r a c t e r i s -
g
t i c r a n g e L h = l / < w > l / p
The a u t h o r whishes t o acknowledge t h e valua- b l e h e l p of D r . G. Kozlowski.
References
/ 2 / De Gennes, P.G., S u p e r c o n d u c t i v i t y of Metals
& A l l o y s (Benjamin 1966)
/ 3 / Saint-James, D., Sarma, G.,Thomas, E.J.,Type I1 Superconductivity(Pergamon Press:OxEord 1969)' 141 Brandt,E.H., J.Low Temp.Phys.
2
(1977)709-735 151 B r a n d t , E.H.,J.Low Temp.Phys.28
(1977)263-291 161 Kogan, V.G., J.Low Temp.Phys. 20 (1975) 103 171 Campbell, A.M., E v e t t s , J.E.,Adv.in Phys.,90
(1972) 199.
181 Campbell, A.M., 1 n t . D i s c u s s i o n Meeting on F l u x P i n n i n g i n Superconductors ed. P.Haasen, H.C., F r e y h a r d t (Goltze, G o t t i n g e n 1975).
191 B r a n d t , E.H., P h i l . Mag. ( t o b e p u b l i s h e d ) . / l o / Dupart, J . M . , B a i x e r a s , J . , F o u r n e t , G.J. Low
