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METASTABLE SUPERCONDUCTIVITY :
MAGNETIC FIELD TRANSVERSE TO A THIN
CYLINDER AND FIELD DEPENDENT
PENETRATION
M. Esfandiari, H. Fink
To cite this version:
METASTABLE SUPERCONDUCTIVITY : MAGNETIC FIELD TRANSVERSE TO A THIN CYLINDER
AND FIELD DEPENDENT PENETRATION
M.R. Esfandiari and H.J. Fink*
Atomic Energy Organization of Iran, Nuclear Research Center, P.O. Box SS27, Teheran, Iran
^Department of Electrical Engineering University of California, Davis, CA 95616
Abstract.- The Ginzburg-Landau (GL) equations for a thin cylinder in a transverse magnetic field were solved for various K-values and radii a and the largest magnetic field, the superheating field Hsjj, was found. Hg^/H (T) has a minimum as a function of a/X(T) at about 1.12HC at a radius a = 3.8X(T) for K-values less than 0.16 \\ = GL penetration depth]- It is found that the Landau critical ppint is at a/X ^ 1.7 only one critical field exists (se-cond order phase transition field), whereas for a/X > 1.7 three fields exist, one being a second (supercooling) and two being first order phase transition (superheating and thermo-dynamic) fields. A general proof is given for the semi-infinite half-space stating that at the superheating field; the magnetic field dependent penetration depth 5(H) = /2X for K-values smaller than 1.1, taking account of fluctuations.
CYLINDER IN A TRANSVERSE MAGNETIC FIELD.- We assume that the z-direction is parallel to the axis of the cylinder, that the applied magnetic field is in the negative x-direction (4> = ir) , and that the cylinder
is very long so that end effects can be neglected. The modulus of the order parameter F(r,(j>) and the superfluid velocity Q(r,<j>) are independent functions of r and ^(cylindrical coordinates) and, by symme-try, assumed to be independent of the z-coordinate. When Q is substituted for Q (r,<t>), the GL equations /!/ in the notation of reference /2/ are
The GL parameter K = X(T)/?(T), where X(T) is the low field penetration depth and 5(T) is the cohe-rence length. The boundary conditions are
3F(r,<(.)/3r| r = a= 0, ( K r , * ) ^ = Q = 0, (3) Q(a,(f>) = -aHa(l-C/a2)sin(j> , (4)
where equation (4) is obtained from potential theo-ry. C is an unknown parameter which is to be
deter-mined by matching the tangential component of the magnetic field at the boundary r = a. By symmetry one needs to solve equations (1) and (2) in the first quadrant only.
Figure 1 shows typical solutions of F(r,<)i) and Q(r,(j)) at the maximum field H . All parameters in this figure are in GL normalized units. The H values are summarized in Figure 2 for different cylinder radii a. This is similar to the case of an applied field parallel to the axis of a cylin-der /2,3/ . F(a,<)>) approaches zero at the maximum field for values of a ^ 1.7X while for a > 1.7X the value of F(a,<|>) is finite at the maximum field. The former is a second order phase transition and the latter a first order phase transition at H
max The curves in Figure 2 approach a constant field value for large values of a/X. From the solutions of the semi-infinite half-space, discussed else-where /3/, it is estimated that the H curve for
0
K = 0.1 approaches the bulk of H ,=1.42H for radii a > 45X ; for K = 0.03 it approaches the bulk value of H . = 2.40 H for a > 150X; and for
sh c 'h '
K = 0.001, H. approaches the bulk value H =i3.35H
0 r r sh c
JOURNAL DE PHYSIQUE
Colloque
C6,
supplément au n"
8,
Tome
39,
août
1978,
page
C6-660
Résumé.-On a résolu les équations de Ginsburg-Landau (GL) relatives à un cylindre mince dans un champ magnétique transversal pour diverses valeurs de K et du rayon a et de plus grand champ. On a trouvé le champ de surchauffe Hsh. Le rapport Hsjj/H (T) en fonction du rapport a A ( T ) présente un minimum pour environ 1,12 Hc à un rayon a = 3,8 X(T) pour des
valeurs de K inférieures à 0.16 [X = profondeur de pénétration de GL] . On trouve que le point critique de Landau est à a/X - 1,7 pour des champs perpendiculaires à l'axe du cy-lindre. Pour a/X % 1,7 seul le champ critique de la transition du second ordre subsiste alors que pour a/X > 1,7 trois champs existent, l'un correspondant à la transition du second ordre (sous refroidissement), les deux autres au premier ordre (surchauffe et ther-modynamique) . On montre que pour le dimi-espace, dans le champ de surchauffe, la profondeur de pénétration dépendant du champ magnétique est, pour les valeurs de K plus petites que
1,1, égale à /2X en tenant compte des fluctuations.
f o r a 4500X.
MAGNETIC FIELD-DEPENDENT PENETRATION DEPTH.-The boundary between normal and superconducting r e g i o n s
i s assumed a t x = 0. The space f o r x > 0 i s f i l l e d
w i t h a superconductor and t h e magnetic f i e l d i s
p a r a l l e l t o t h e p o s i t i v e z - d i r e c t i o n .
-
0 0 1 0.4 0.6 0.8 1.02+/* ; r i a
F i g . 1 : S o l u t i o n s of t h e % w a i T m e t e e ?
2%
s u p e r f l u i d v e l o c i t y a s a f u n c t i o n of r a d i u s a and a n g l e $ a r e shown i n t h e f i r s t quadrant (cylinde- r i c a l c o o r d i n a t e s ) f o r a superconducting c y l i n d e r i n a t r a n s v e r s e magnetic f i e l d . The p l o t i s f o r
K = 0.03 and a = 7. A l l parameters a r e i n GL nor- malized u n i t .
F i g . 2 : Comparison of t h e o r e t i c a l and experimental r e s u l t s of s u p e r h e a t i n g and supercooling of a c y l i n d e r i n a t r a n s v e r s e magnetic f i e l d . Experimen- t a l p o i n t s a r e those of indium c r y s t a l whiskers/lO/ whose d i a m e t e r s a r e 4.5 pm ( s o l i d o i n t s ) and 2.8
vrn
(open c i r c l e s ) . X(0) = 5201.
Theory : S o l i d l i n e s . Since H(x) = dQ/dx, t h e magnetic f i e l d dependent p e n e t r a t i o n depth 6 i s6
-
1
(H(x)/H(O))dx-
-P(O)/H(O). (5) 0 One o b t a i n s /3/ from t h e f i r s t i n t e e r a l of t h e GL e q u a t i o n s a t t h e f r e e s u r f a c e of t h e superconductora r e l a t i o n between H(0): H and 6 which i s
0
It i s known from s o l u t i o n s f o r t h e s e m i - i n f i n i t e half-space 141 t h a t a t x
=
0~ H @ , F ( o ~ / ~ F ( o ) = 0 (7) a t t h e maximum f i e l d . When a v a r i a t i o n a l c a l c u l a - t i o n on t h e f r e e energy i s performed i n t h e manner of t h e s t i b i l i t y a n a l y s i s by Fink and Presson /4/ a t t h e maximum f i e l d a t which e q u a t i o n (7) a p p l i e s , one f i n d s t h a t t h e second v a r i a t i o n
6Zg
i s alway's zero. From t h i s , one i s n o t a b l e t o e x t r a c t i n f o r - mation concerning t h e f i e l d dependent p e n e t r a t i o n d e p t h 6. However, from t h e t h i r d v a r i a t i o ng3g
= 0 one f i n d s a t t h e maximum f i e l d From e q u a t i o n s (6) and (8) t h e s u r p r i s i n g r e s u l t emerges t h a t a t t h e maximum f i e l d , r e g a r d l e s s of v a l u e of K , 6th=a,
(9)The f o u r t h v a r i a t i o n of t h e Gibbs f r e e energy i s p o s i t i v e d e f i n i t e . Equation (9) i s v a l i d f o r a l l K-values a t t h e maximum f i e l d , which f o r K < 1 . 1 i s
t h e s u p e r h e a t i n g f i e l d . For K > ] . I . , t h e superhea- t i n g f i e l d i s t h e o r e t i c a l l y s m a l l e r t h a n t h e maxi- mum f i e l d due t o i n s t a b i l i t i e s / 4 / .
COMPARISON WITH EXPERIMENTS.- I n o r d e r t o o b t a i n experimental curves r e l a t e d t o Figure 2 , one ought t o v a r y t h e r a t i o n a/X(T) and measure Hsh(T)/Hc(T). One way t o do t h i s i s t o v a r y t h e temperature of a superconducting c y l i n d e r of c o n s t a n t r a d i u s a and measure t h e change i n a/X(T) due t o a change i n t h e p e n e t r a t i o n depth X(T) which v a r i e s from a f i n i t e v a l u e a t T = 0 t o i n f i n i t y a t T = Tc.
F i g u r e 2 shows t h e s u p e r h e a t i n g and super- c o o l i n g f i e l d s f o r d i f f e r e n t :-values f o r a c y l i n - d e r i n a t r a n s v e r s e magnetic f i e l d . The experimen- t a l p o i n t s a r e measurements on indium microcylin- d e r s by Michael and McLachlan
151.
I n / 5 / t h e K-values a r e p l o t t e d a s a f u n c t i o n of t h e reduced temperature. We have r e c o n s t r u c t e d t h e s e p o i n t s i n Figure 2 with t h e a i d of h ( t )-
X(0)/2(1-t)1/2. The l i k e l i h o o d of t h e minimum i n Hsh/Hc a s a func- t i o n of a/X(T) i s apparent from t h i s f i g u r e .Recent measurements by P a r r /6/ show t h a t
References
/I/ Ginzburg, V.L. and Landau, L.D., Zh. Eksp. i. Theor. Fiz.
2
(1950) 1064. /2/ Esfandiari, M.R. and Fink, H . J . , Phys. Letters54A
(1975) 383./3/ Esfandiari, M.R., Thesis, Univ. Of Calif., Davis, 1977, unpublished. /4/ Findk, H . J . and Presson, A.G., Phys. Rev.
182
(1969) 498./ 5 / Michael, P. and McLachlan, D.S., J . Low Temp. Phys
14
(1974) 604./6/ Parr, H . , Phys. Rev. B g (1976) 2842; Phys. Rev. B
