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Angular dependence of the pinning and the flow of vortices in thin superconducting films
Volume 23, number 3
P HYSI C S L E T T E R S
17 October 1966
ANGULAR DEPENDENCE OF THE PINNING AND THE FLOW OF VORTICES IN THIN SUPERCONDUCTING FILMS * R. D E L T O U R ** and M. TINKHAM
University of California, Berkeley, California, USA Received 27 September 1966 The pinning force and the flow regime of vortices in thin superconducting films are shown to depend essentialy on the perpendicular component of the magnetic field with respect to the plane of the films.
The e f f e c t of changing the angle b e t w e e n a constant m a g n e t i c f i el d H a n d a d.c. c u r r e n t 1 flowing in a type H s u p e r c o n d u c t o r h a s been i n v e s t i g a t e d in w i r e s and thick foil g e o m e t r i e s by v a r i o u s a u t h o r s [1,2]. T h e i r r e s u l t s a r e c o n s i s t e n t with a L o r e n t z f o r c e type m e c h a n i s m b e ing r e s p o n s i b l e f o r the motion of the v o r t i c e s in the m a t e r i a l . Swartz and H a r t [2] have a l s o studied the dependence of the c r i t i c a l c u r r e n t on the m a g n e t i c f i e l d and its angle with the s u r f a c e of a ribbon of type II m a t e r i a l and have i n f e r r e d the e x i s t e n c e of two d i f f e r e n t m e c h a n i s m s c o n t r i b u t i n g to the t r a n s p o r t of c u r r e n t in t h e s e geometries. We have i n v e s t i g a t e d the influence of the mutual o r i e n t a t i o n of the magnetic field, the c u r r e n t , and the s u r f a c e on the r e s i s t i v e t r a n s i t i o n of s u p e r c o n d u c t i n g f i l m s e v a p o r a t e d on a g l a s s s u b s t r a t e . The f i l m s , a p p r o x i m a t e l y 2 m m wide and 10 m m long, w e r e i m m e r s e d in a liquid 4He bath whose v a p o r p r e s s u r e w a s continuously r e g u l a t e d . The longitudinal voltage a c r o s s the f i l m w a s d i r e c t l y m e a s u r e d u s in g a 4 t e r m i n a l t e c hni qu e on a Hewlett P a c k a r d 419B v o l t m e t e r conne c ted to an X Y r e c o r d e r . The t h i c k n e s s of the f i l m s (tin) w a s i n f e r r e d f r o m m e a s u r e d v a l u e s of the p a r a l l e l and p e r p e n d i c u l a r c r i t i c a l f i e l d s Hc, , and Hcj" [3]. Fig. 1 r e p r e s e n t s the e x p e r i m e n t a l v a l u e s of the c r i t i c a l f i e l d s H c r at which t h e r e a p p e a r s a longitudinal v o l t ag e (>~ 0.2 ~tV) along the f i l m c a r r y i n g a f i x ed d.c. c u r r e n t , when v a r y i n g the angle b e t w e e n the m a g n e t i c f i e l d and the plane of the f i l m , both in the plane p e r p e n d i c u l a r to I and in the plane c o n t a i n i n g / . Note that in the s e t of * Research supported in part by N.S.F and O.N.R. ** Charge de Recherches du F.N.R.S. (Belgium). Present address: Gordon McKay Lab., Harvard Univ., Cambridge,
Mass.,
USA.
IC -
~
H
HCr (~) Her (~'/2 )
5
0
I
30
I
60
I ~,
90 8 (DEGREES)
Fig. 1. Angular dependence of Her, the field for first appearance of resistance for a 1 mA d.c. current in two thin films of tin (thickness 420 ~) at T = 0.95 Tc. a) o: go= ½1r; H c i = 40 gauss, Hc~j --- 950 gauss, Hcr(½1r) = 10.5 gauss. h) x: go= 0; Hc_L= 49 gauss, Hc, = 1060 gauss, Hcr(½~) = 22 gauss. e x p e r i m e n t s with go = ~ r , / - / a n d / a r e always p e r p e n d i c u l a r to e a c h o t h er w h e r e a s when go = 0, the angle b e t w e e n H and I is d i f f e r e n t f o r each e x p e r iment. F o r thin f i l m s (d << ~T) the solution of the G i n z b u r g - L a n d a u equations in the p r e s e n c e of a m a g n e t i c f i el d shows that g/(r) is e s s e n t i a l l y only a function of the two i n - p l a n e c o o r d i n a t e s . The p e r p e n d i c u l a r c o m p o n e n t of the m a g n e t i c f i el d d e t e r m i n e s the v o r t e x s t r u c t u r e in the f i l m [3] while the p a r a l l e l component only a f f e c t s the magnitude of ~(r) through a s c a l e f a c t o r . T h e r e 183
Volume 23, n u m b e r 3
PHYSICS
LETTERS
f o r e , t h e c r i t i c a l f i e l d H c r ( e ) , c o r r e s p o n d i n g to t h e f i r s t a p p e a r a n c e of a r e s i s t i v e b e h a v i o r , measured for the field at any angle 0 with the p l a n e of t h e f i l m i s e x p e c t e d to b e e s s e n t i a l l y d e t e r m i n e d b y t h e n o r m a l c o m p o n e n t of t h e f i e l d leading to the relation: Hcr(0) sin 0 = Hcr(~ ) .
e b
8 80 80
H 2 3
d e
80 80
5 6
¢ 80 4
V
(1)
The corresponding theoretical curve is repres e n t e d a s a c o n t i n u o u s l i n e i n fig. 1. T h e good a g r e e m e n t b e t w e e n (1) a n d t h e e x perimental points up to very small angles indic a t e s t h a t t h e p a r a l l e l c o m p o n e n t s of t h e f i e l d i n d e e d h a v e l i t t l e e f f e c t o n t h e d e p i n n i n g of t h e perpendicular component. This is reasonzble, since even for the small angle points, H, is considerably smaller than Hc,. Thus H, should not g r e a t l y r e d u c e t h e m a g n i t u d e of t h e s u p e r c o n ducting order parameter, nor, consequently, the pinning energy. In fig. 2, we h a v e v e r i f i e d t h a t t h e p e r p e n d i c ular component of/t also plays the dominant role in the vortex-flow regime. The continuous curves are the experimental V versus I dependence for a p a r t i c u l a r o r i e n t a t i o n (~ = 80 ° , ~0 = 90 °) a n d t h e dashed curves represent the experimental V v e r s u s I c u r v e s o b t a i n e d a t other 0 a n g l e s i n a m a g n e t i c f i e l d of s u c h s t r e n g t h a s to h a v e n e a r l y t h e same perpendicular component. W e n o t e a g a i n t h e a p p a r e n t l a c k of i n f l u e n c e of t h e f i e l d c o m p o n e n t p a r a l l e l to t h e f i l m p l a n e , e v e n w h e n it greatly exceeds the perpendicular component. In f a c t , the i n f l u e n c e of H,, w a s m a n i f e s t o n l y f o r v e r y s m a l l a n g l e s , w h e r e it o b s e r v a b l y i n c r e a s e d t h e f l u x f l o w v o l t a g e f o r a g i v e n I a n d H±. A s a n e x a m p l e , a c u r v e e s s e n t i a l l y i d e n t i c a l to d w a s obtained at an angle 8 = 1o for a field whose perp e n d i c u l a r c o m p o n e n t w a s o n l y a b o u t ½ of t h e v a l u e of H . c o r r e s p o n d i n g t o t h e s a m e c u r v e o b t a i n e d a t e = 12.5 ° o r 80 ° . W e w o u l d l i k e to t h a n k F . H a r p e r f o r t h e u s e of h i s c r y o s t a t d u r i n g t h e c o m p l e t i o n of t h i s w o r k . O n e of u s (R.D.) i s v e r y g r a t e f u l to P r o f e s s o r * * * * *
184
17 October 1966 8' H' tsinOtt'sine 12.5 8 1197 1.73 7.5 20 2.96 2.6t 40 6 3.94 3.87 25 t2 4.93 5.06 12.5 28 5.92 6.05
7/ 500~V
/¢
//
/ T e / /
,/I ? 0 0
~ lOOmA
=I
Fig. 2. Angular dependence of the flux flow r e g i m e for a film of tin (thickness 1600 ~) at T = 0.96 Tc (Hc± = 15 gauss, Hcl r = 150 gauss). The solid lines c o r r e s p o n d to the u n p r i m e d values of e and H shown in the table on top of the figure whereas the dashed lines c o r r e s p o n d to the p r i m e d values, a ' and H' (respectively m e a s u r e d in d e g r e e s and gauss). A. F . Kip w h o h e l p e d m u c h t o m a k e h i s s t a y a t Berkeley possible and to the Belgium NATO s c i e n t i f i c c o m m i t t e e f o r t h e g r a n t of a r e s e a r c h fellowship.
References 1. C. F. Hempstead and Y. B. Kim, Phys. Rev. L e t t e r s 12 (1964) 145; W. F. Druyvesteyn and J. Volger, Philips Res. Repts. 19 (1964) 359; S. J. Williamson and J. K. Furdyna, Phys. L e t t e r s 21 (1966) 376. 2. P.S. Swartz and H. R. Hart, Phys. Rev. 137 (1965) A818. 3. M. Tinkham, Phys. Rev. 129 (1963) 2413.
P HYSI C S L E T T E R S
17 October 1966
ANGULAR DEPENDENCE OF THE PINNING AND THE FLOW OF VORTICES IN THIN SUPERCONDUCTING FILMS * R. D E L T O U R ** and M. TINKHAM
University of California, Berkeley, California, USA Received 27 September 1966 The pinning force and the flow regime of vortices in thin superconducting films are shown to depend essentialy on the perpendicular component of the magnetic field with respect to the plane of the films.
The e f f e c t of changing the angle b e t w e e n a constant m a g n e t i c f i el d H a n d a d.c. c u r r e n t 1 flowing in a type H s u p e r c o n d u c t o r h a s been i n v e s t i g a t e d in w i r e s and thick foil g e o m e t r i e s by v a r i o u s a u t h o r s [1,2]. T h e i r r e s u l t s a r e c o n s i s t e n t with a L o r e n t z f o r c e type m e c h a n i s m b e ing r e s p o n s i b l e f o r the motion of the v o r t i c e s in the m a t e r i a l . Swartz and H a r t [2] have a l s o studied the dependence of the c r i t i c a l c u r r e n t on the m a g n e t i c f i e l d and its angle with the s u r f a c e of a ribbon of type II m a t e r i a l and have i n f e r r e d the e x i s t e n c e of two d i f f e r e n t m e c h a n i s m s c o n t r i b u t i n g to the t r a n s p o r t of c u r r e n t in t h e s e geometries. We have i n v e s t i g a t e d the influence of the mutual o r i e n t a t i o n of the magnetic field, the c u r r e n t , and the s u r f a c e on the r e s i s t i v e t r a n s i t i o n of s u p e r c o n d u c t i n g f i l m s e v a p o r a t e d on a g l a s s s u b s t r a t e . The f i l m s , a p p r o x i m a t e l y 2 m m wide and 10 m m long, w e r e i m m e r s e d in a liquid 4He bath whose v a p o r p r e s s u r e w a s continuously r e g u l a t e d . The longitudinal voltage a c r o s s the f i l m w a s d i r e c t l y m e a s u r e d u s in g a 4 t e r m i n a l t e c hni qu e on a Hewlett P a c k a r d 419B v o l t m e t e r conne c ted to an X Y r e c o r d e r . The t h i c k n e s s of the f i l m s (tin) w a s i n f e r r e d f r o m m e a s u r e d v a l u e s of the p a r a l l e l and p e r p e n d i c u l a r c r i t i c a l f i e l d s Hc, , and Hcj" [3]. Fig. 1 r e p r e s e n t s the e x p e r i m e n t a l v a l u e s of the c r i t i c a l f i e l d s H c r at which t h e r e a p p e a r s a longitudinal v o l t ag e (>~ 0.2 ~tV) along the f i l m c a r r y i n g a f i x ed d.c. c u r r e n t , when v a r y i n g the angle b e t w e e n the m a g n e t i c f i e l d and the plane of the f i l m , both in the plane p e r p e n d i c u l a r to I and in the plane c o n t a i n i n g / . Note that in the s e t of * Research supported in part by N.S.F and O.N.R. ** Charge de Recherches du F.N.R.S. (Belgium). Present address: Gordon McKay Lab., Harvard Univ., Cambridge,
Mass.,
USA.
IC -
~
H
HCr (~) Her (~'/2 )
5
0
I
30
I
60
I ~,
90 8 (DEGREES)
Fig. 1. Angular dependence of Her, the field for first appearance of resistance for a 1 mA d.c. current in two thin films of tin (thickness 420 ~) at T = 0.95 Tc. a) o: go= ½1r; H c i = 40 gauss, Hc~j --- 950 gauss, Hcr(½1r) = 10.5 gauss. h) x: go= 0; Hc_L= 49 gauss, Hc, = 1060 gauss, Hcr(½~) = 22 gauss. e x p e r i m e n t s with go = ~ r , / - / a n d / a r e always p e r p e n d i c u l a r to e a c h o t h er w h e r e a s when go = 0, the angle b e t w e e n H and I is d i f f e r e n t f o r each e x p e r iment. F o r thin f i l m s (d << ~T) the solution of the G i n z b u r g - L a n d a u equations in the p r e s e n c e of a m a g n e t i c f i el d shows that g/(r) is e s s e n t i a l l y only a function of the two i n - p l a n e c o o r d i n a t e s . The p e r p e n d i c u l a r c o m p o n e n t of the m a g n e t i c f i el d d e t e r m i n e s the v o r t e x s t r u c t u r e in the f i l m [3] while the p a r a l l e l component only a f f e c t s the magnitude of ~(r) through a s c a l e f a c t o r . T h e r e 183
Volume 23, n u m b e r 3
PHYSICS
LETTERS
f o r e , t h e c r i t i c a l f i e l d H c r ( e ) , c o r r e s p o n d i n g to t h e f i r s t a p p e a r a n c e of a r e s i s t i v e b e h a v i o r , measured for the field at any angle 0 with the p l a n e of t h e f i l m i s e x p e c t e d to b e e s s e n t i a l l y d e t e r m i n e d b y t h e n o r m a l c o m p o n e n t of t h e f i e l d leading to the relation: Hcr(0) sin 0 = Hcr(~ ) .
e b
8 80 80
H 2 3
d e
80 80
5 6
¢ 80 4
V
(1)
The corresponding theoretical curve is repres e n t e d a s a c o n t i n u o u s l i n e i n fig. 1. T h e good a g r e e m e n t b e t w e e n (1) a n d t h e e x perimental points up to very small angles indic a t e s t h a t t h e p a r a l l e l c o m p o n e n t s of t h e f i e l d i n d e e d h a v e l i t t l e e f f e c t o n t h e d e p i n n i n g of t h e perpendicular component. This is reasonzble, since even for the small angle points, H, is considerably smaller than Hc,. Thus H, should not g r e a t l y r e d u c e t h e m a g n i t u d e of t h e s u p e r c o n ducting order parameter, nor, consequently, the pinning energy. In fig. 2, we h a v e v e r i f i e d t h a t t h e p e r p e n d i c ular component of/t also plays the dominant role in the vortex-flow regime. The continuous curves are the experimental V versus I dependence for a p a r t i c u l a r o r i e n t a t i o n (~ = 80 ° , ~0 = 90 °) a n d t h e dashed curves represent the experimental V v e r s u s I c u r v e s o b t a i n e d a t other 0 a n g l e s i n a m a g n e t i c f i e l d of s u c h s t r e n g t h a s to h a v e n e a r l y t h e same perpendicular component. W e n o t e a g a i n t h e a p p a r e n t l a c k of i n f l u e n c e of t h e f i e l d c o m p o n e n t p a r a l l e l to t h e f i l m p l a n e , e v e n w h e n it greatly exceeds the perpendicular component. In f a c t , the i n f l u e n c e of H,, w a s m a n i f e s t o n l y f o r v e r y s m a l l a n g l e s , w h e r e it o b s e r v a b l y i n c r e a s e d t h e f l u x f l o w v o l t a g e f o r a g i v e n I a n d H±. A s a n e x a m p l e , a c u r v e e s s e n t i a l l y i d e n t i c a l to d w a s obtained at an angle 8 = 1o for a field whose perp e n d i c u l a r c o m p o n e n t w a s o n l y a b o u t ½ of t h e v a l u e of H . c o r r e s p o n d i n g t o t h e s a m e c u r v e o b t a i n e d a t e = 12.5 ° o r 80 ° . W e w o u l d l i k e to t h a n k F . H a r p e r f o r t h e u s e of h i s c r y o s t a t d u r i n g t h e c o m p l e t i o n of t h i s w o r k . O n e of u s (R.D.) i s v e r y g r a t e f u l to P r o f e s s o r * * * * *
184
17 October 1966 8' H' tsinOtt'sine 12.5 8 1197 1.73 7.5 20 2.96 2.6t 40 6 3.94 3.87 25 t2 4.93 5.06 12.5 28 5.92 6.05
7/ 500~V
/¢
//
/ T e / /
,/I ? 0 0
~ lOOmA
=I
Fig. 2. Angular dependence of the flux flow r e g i m e for a film of tin (thickness 1600 ~) at T = 0.96 Tc (Hc± = 15 gauss, Hcl r = 150 gauss). The solid lines c o r r e s p o n d to the u n p r i m e d values of e and H shown in the table on top of the figure whereas the dashed lines c o r r e s p o n d to the p r i m e d values, a ' and H' (respectively m e a s u r e d in d e g r e e s and gauss). A. F . Kip w h o h e l p e d m u c h t o m a k e h i s s t a y a t Berkeley possible and to the Belgium NATO s c i e n t i f i c c o m m i t t e e f o r t h e g r a n t of a r e s e a r c h fellowship.
References 1. C. F. Hempstead and Y. B. Kim, Phys. Rev. L e t t e r s 12 (1964) 145; W. F. Druyvesteyn and J. Volger, Philips Res. Repts. 19 (1964) 359; S. J. Williamson and J. K. Furdyna, Phys. L e t t e r s 21 (1966) 376. 2. P.S. Swartz and H. R. Hart, Phys. Rev. 137 (1965) A818. 3. M. Tinkham, Phys. Rev. 129 (1963) 2413.