On the concept of pinning force in type II superconductors

Volume 25A. number 5

ON THE

PHYSICS

CONCEPT

OF

PINNING

LETTERS

FORCE

11 September 1967

IN TYPE

II SUPERCONDUCTORS

K. Y A M A F U J I and F . IRIE Department of Electronics. Kyushu University, Fukuoka, Japan Received 3 August 1967

It is pointed out that the former derivations of the pinning force in non-ideal type II superconductors are x~Tong in principle. An expression of pinning force is derived in case of a linear fluxoid-lattiee, as an example. The concept of "pinning f o r c e " w a s f i r s t i n t r o d u c e d by Kim et al. [1] in an equation of f o r c e b a l a n c e f o r the flux flow in a t y p e II s u p e r c o n ductor as f L - f v =fp

(1)

w h e r e f L , f v , a n d f p a r e the L o r e n t z f o r c e , v i s c o u s r e s i s t i v e - f o r c e , and pinning f o r c e , r e s p e c t i v e l y , a c t i n g on a fluxoid p e r unit length. The o r i g i n of the pinning f o r c e h a s b e e n a t t r i b u t e d to f l u x o i d - p i n i n t e r a c t i o n s . A t y p i c a l d e r i v a t i o n of the pinning f o r c e h a s b e e n c a r r i e d out [9.] by c a l c u l a t i n g the p o t e n t i a l , Upf, due to the i n t e r a c t i o n b e t w e e n a s i n g l e fluxoid-and a pin s o a s to obtain the s t a t i c t h r e s h o l d f o r c e , f p f = -(dUuf/dX)ma xSuch a d e r i v a t i o n i s , howe- v e r , wro" rig in p r i n ciple. F i r s t , eq. (1) r e p r e s e n t s the f o r c e b a l a n c e when f l u x o i d s a r e in motion. If we m u l t i p l y the flow v e l o c i t y ~ on the both hand s i d e s of eq. (1), we can s e e t h a t the pinning f o r c e should p r o d u c e a k i n d of r e s i s t i v e l o s s b y the m o t i o n of fluxoids. Such a d y n a m i c pinning f o r c e d o e s not coincide with the s t a t i c t h r e s h o l d f o r c e in g e n e r a l . Second, the l a t t i c e e n e r g y of the A b r i k o s o v fluxoids h a s not b e e n t a k e n into account in the f o r m e r d e r i v a t i o n s of pinning f o r c e . Since the m a c r o s c o p i c r e s i s t i v e - f o r c e due to p i n s c a n b e c a l c u l a t e d b y c o n s i d e r i n g the s c a t t e r i n g of moving fluxoids due to p i n s , the l o c a l d e f o r m a t i o n e n e r g y of fluxoidl a t t i c e should give an e s s e n t i a l c o n t r i b u t i o n to the pinning f o r c e . T h i r d , e v e r y f l u x o i d cannot i n t e r a c t d i r e c t l y with p i n s in g e n e r a l . The p i n ning f o r c e a p p e a r s in eq. (1) i s an a v e r a g e d f o r c e , which i s not equal to the pinning f o r c e a c t ing on a given f l u x o i d d i r e c t l y i n t e r a c t i n g with a pin. Since the t h i r d point w a s d i s c u s s e d in o u r r e cent p a p e r [3], we s h a l l d i s c u s s the f i r s t two p o i n t s h e r e . F o r s i m p l i c i t y , let u s c o n s i d e r a l i n e a r l a t t i c e and s u p p o s e that the f l u x o i d - l a t t i c e

i s m a k i n g s t e a d y flow in a m a c r o s c o p i c s e n s e . The f l u x o i d - l a t t i c e s u f f e r s a s m a l l d e f o r m a t i o n in the v i c i n i t y of e a c h pin, and we a s s u m e that t h e r e s t o r i n g f o r c e due to such a d e f o r m a t i o n i s o b e y e d by Hooke*s law. Then the equation of m o tion f o r a fluxoid p e r unit length i s given by MA~+ 7/~ + kAx = -dUpf/dx,

(2)

w h e r e a x = x - ~ i s the d e v i a t i o n of t h e p o s i t i o n of the fluxoid f r o m i t s m e a n p o s i t i o n ~ i~. the s t e a d y flow with a c o n s t a n t v e l o c i t y v = x, M the e f f e c t i v e m a s s of the fluxoid p e r unit length, 77 t h e v i s c o s i t y c o e f f i c i e n t , and k the r e s t o r i n g f o r c e c o n s t a n t due to d e f o r m a t i o n of the fluxoidlattice. L e t u s c o n s i d e r the e n e r g y l o s s d u r i n g the m o tion of the fluxoid a c r o s s a given p i n (cf. fig. 1). I n t e g r a t i o n of eq. (2) u n d e r the c y c l i c b o u n d a r y c o n d i t i o n s at t = 0 and t = ~o l e a d s to 1 tp

--

fo

dt.

iS)

Since the t o t a l l o s s p o w e r , 7/v -'2, should b e equal to the input p o w e r , f L ~, the c o m p a r i s o n of eq. (3) l e a d s to f p = f L " BY = ( - k A x ) .

(4)

T h e m e a n i n g of the p r e s e n t pinning f o r c e i s a s follows. T h e d e f o r m a t i o n e n e r g y of the f l u x o t d -

~

f

V Fig. 1. 387

Volume25A. number 5

PHYSICS

l a t t i c e r i s e s l o c a l l y due to the f l u x o i d - p i n i n t e r a c t i o n p o t e n t i a l . Such a d e f o r m a t i o n e n e r g y t r a n s f e r s to the v i b r a t i o n e n e r g y of the f l u x o i d - l a t t i c e d u r i n g the m o t i o n of f l u x o i d s a c r o s s the given pin, and a p a r t of the v i b r a t i o n e n e r g y d i s s i p a t e s d u r i n g the m o t i o n of f l u x o i d s in the v i s c o u s m e dium. Such a d i s s i p a t e d e n e r g y d o e s give the so c a l l e d pinning l o s s e n e r g y . To show an e s s e n t i a l f e a t u r e of such a pinning f o r c e , let u s s i m p l i f y the p o t e n t i a l a s ldUof/dx t = = const. + f p f (cf. the dotted l i n e s in fig. 1), and f u r t h e r a s s u m e the v i b r a t i o n d a m p s in a v e r y s h o r t t i m e . In such a l i m i t i n g c a s e , a l l of the l o c a l d e f o r m a t i o n e n e r g y t r a n s f e r s to the pinning l o s s and eq. (4) l e a d s to

11 September 1967

the d e f o r m a t i o n e n e r g y of f l u x o i d - l a t t i c e , a s has b e e n done in f o r m e r d e r i v a t i o n s of fu, the pinning f o r c e d o e s not a p p e a r u n l e s s the f l u ~ o i d - p i n i n t e r a c t i o n i s a t t r a c t i v e . Second, the c r i t i c a l c u r r e n t d a t a in the flux flow shows that the p i n ning f o r c e d o e s d e c r e a s e r a p i d l y at the f i e l d s e x t r e m e l y n e a r Hc2 [3]. Since the r e s t o r i n g f o r c e constant k should i n c r e a s e r a p i d l y [4] at t h e s e f i e l d s , such a b e h a v i o r of the pinning f o r c e can b e u n d e r s t o o d at l e a s t q u a l i t a t i v e l y by the p r e s e n t r e s u l t . The r e s u l t of d e t a i l e d c a l c u l a t i o n in a t h r e e d i m e n s i o n a l s a m p l e w i l l be p u b l i s h e d e l s e where.

References

fp : (3fpf/kap pf. The pinning f o r c e in eq. (5) h a s two r e m a r k a b l e f e a t u r e s . F i r s t , the p r e s e n t pinning f o r c e a p p e a r s in both the c a s e s when the f l u x o i d - p i n i n t e r a c t i o n i s a t t r a c t i v e and r e p u l s i v e , b e c a u s e fp fpf2. If we c o n s i d e r a single fluxoid i g n o r i n g

LOW

LETTERS

TEMPERATURE

1. Y.B. Kim, C. F. Hempstead and A. R. Strnad, Phys. Rev. 139 (1963) Al163. 2. W.W. Webb, Phys.Rev. Letters 11 (1963) 191. 3. F . I r i e and K. Yamafuji. J. Phys.Soc. Japan 23, no. 2, to be published. 4. C. Caroli and K. Maki. Phys. Rev. Letters 18 (1967) 698.

SPECIFIC

HEAT

OF

EuSe

*

H. W. WHITE, D. C. McCOLLUM and J. CALLAWAY

Physics Department. University of California. Riverside, California, USA Received 4 July 1967

Specific heat measurements on EuSe from 0.5 to 6°K are reported. An anomaly at 2.8°K was found.

Busch [1] h a s c o n c l u d e d f r o m o p t i c a l and s u s c e p t i b i l i t y d a t a that EuSe in z e r o m a g n e t i c f i e l d i s a n t i f e r r o m a g n e t i c above a p p r o x i m a t e l y 4OK and f e r r o m a g n e t i c below 2.8OK. P r e v i o u s l y r e p o r t e d s p e c i f i c h e a t r e s u l t s [2] down to 1.3OK showed one l a r g e a n o m a l y l o c a t e d at 4 . 6 o i c Since the e n t r o p y d i f f e r e n c e b e t w e e n the f e r r o m a g n e t i c and a n t i f e r r o m a g n e t i c p h a s e s m a y be s m a l l , only a s m a l l a n o m a l y , if any, i s to be e x p e c t e d in the s p e c i f i c heat n e a r the o n s e t of the f e r r o m a g n e t i c a l l y o r d e r e d p h a s e . Specific h e a t m e a s u r e m e n t s * Research sponsored in part by the Air Force Office of Scientific Research. Office of Aerospace Research, U.S.Air Force, under AFOSR Grant No. 523-64. 388

on EuSe have b e e n u n d e r t a k e n to s e a r c h f o r any a n o m a l i e s which might be a t t r i b u t e d to the onset of t h i s new o r d e r i n g and a l s o to i n v e s t i g a t e the t e m p e r a t u r e d e p e n d e n c e down to 0.5oIC M e a s u r e m e n t s w e r e m a d e on t h r e e p r e s s e d p o w d e r s a m p l e s , d e s i g n a t e d by I, H, and HI, weighing 2.53, 1.11, and 0.30 g r e s p e c t i v e l y . S a m p l e I w a s p r e p a r e d by r e a c t i n g e u r o p i u m m e t a l , g r a d e 99.9%, obtained in 1967 f r o m E l e c t r o n i c Space P r o d u c t s , Inc. (ESPI) with high p u r i t y s e l e n i u m , g r a d e 99. 999%, o b t a i n e d f r o m A m e r i c a n S m e l t i n g and Refining Co. in a s e a l e d q u a r t z c o n t a i n e r at 1000oc f o r 12 h o u r s . S a m p l e H, which w a s kindly p r o v i d e d by Mr. R. Brown, w a s p r e p a r e d in a s i m i l a r m a n n e r using s e l e n i u m (99. 999%) and e u r o p i u m (99.9%) obtained f r o m