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The rounding of the resistive transition in metallic lead films
V o l u m e 28A, n u m b e r 3
PHYSICS
LETTERS
w h e r e the s y s t e m i s a s s u m e d to h a v e a f i n i t e n u m b e r of a c c e s s i b l e q u a n t u m s t a t e s w e i g h t e d w i t h p r o b a b i l i t i e s P l , 1 = 1 , . . . , n and is o t h e r w i s e uns p e c i f i e d . A n o t h e r b a s i c a s s u m p t i o n r e q u i r e d in t h e u n i q u e n e s s t h e o r e m i s that S { P l } a t t a i n s a m a x i m u m w i t h r e s p e c t to e a c h P l in a n e q u i l i b r i u m s t a t e . T h e q u a n t u m s t a t i s t i c a l e r g o d i c and H t h e o r e m s , b o t h of w h i c h a r e a x i o m a t i c in n a t u r e , a r e l a t e n t i n g r e d i e n t s in the p r o o f of the u n i q u e n e s s of the f o r m S g i v e n a b o v e . O u r a p p r o x i m a t e k i n e t i c t r e a t m e n t is c l e a r l y l e s s g e n e r a l , but
18 N o v e m b e r 1968
d o e s not s e e m to r e l y on t h e s e t h e o r e m s .
References 1. s. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases (Cambridge at the University P r e s s , 1960). 2. L.D. Landau and E. M. Lifshitz, Fluid mechanics (Pergamon P r e s s , 1959) p. 187. 3. P . T . Landsberg, Thermodynamics (Interscience, 1961) Appendix C.
****~
THE
ROUNDING OF THE IN METALLIC
RESISTIVE TRANSITION LEAD FILMS *
R. O. SMITH, B. S E R I N and E. A B R A H A M S Department of Physics, Rutgers University, New Brunswick, NJ. R e c e i v e d 3 S e p t e m b e r 1968
The resistive transition of metallic lead films exhibit the rounding predicted by mean field theories of thermal fluctuations. Close to Tc, the data depend on current density; deviations from the predicted behavior as T--- T c are consistent with the expected breakdown of the theory.
We h a v e i n v e s t i g a t e d the r e s i s t i v e t r a n s i t i o n of m e t a l l i c l e a d f i l m s and h a v e found a r o u n d i n g of the r e s i s t a n c e w h i c h a g r e e s w e l l with t h e o r e t i c a l p r e d i c t i o n s [1,2]. T h e r e s i s t a n c e of a f i l m w i t h t h i c k n e s s d i s p r e d i c t e d to v a r y w i t h t e m p e r a t u r e a c c o r d i n g to RN/R(T)
- 1 = TO/~"
(1)
w h e r e "r = (T - T c ) / T c , TO = R r e s e 2 / ( 1 6 ~ ) and R r e s = 1/(oNd)f~. T h e r a t i o ~ ' o / R r e s is then t h e s a m e f o r a l l f i l m s and e q u a l to 0.152 × 10 - 4 ~ - 1 . W h i l e s o m e p r e v i o u s e x p e r i m e n t a l d a t a [3] on a m o r p h o u s s t r u c t u r e s and v e r y d i r t y s y s t e m s show reasonable agreement with this result, other r e s u l t s [4] a r e u n c l e a r o r d i s a g r e e [5]. H o w e v e r , our results for ~-o/Rres for metallic lead films a r e v e r y c l o s e to t h e t h e o r e t i c a l v a l u e . W e a l s o find that a s ~- ~ 0 d e p a r t u r e s f r o m eq. (1) o c c u r at t e m p e r a t u r e s f o r w h i c h the s u p e r f l u i d v e l o c i t y i s of the o r d e r of a c r i t i c a l v e l o c i t y . * Research supported by the U. S. Army Research and Development Laboratory, Fort Monmounth, N . J . , the National Science Foundation, and the Rutgers Research Council. 224
F i l m s of v a r i o u s t h i c k n e s s e s s w e r e c o n d e n s ed onto a thin g l a s s s u b s t r a t e h e l d at t e m p e r a t u r e s r a n g i n g f r o m 6 to 10OK. T h i s p r o v i d e d r e a s o n a b l e v a r i a t i o n s in the r e s i d u a l r e s i s t a n c e s , R r e s , with w h i c h to t e s t the c a l c u l a t i o n s . T h e c o r r e s p o n d i n g m e a n f r e e p a t h s v a r i e d f r o m 15 to 40 A. T h e t r a n s i t i o n w i d t h s , d e f i n e d by the t e m p e r a t u r e r a n g e in w h i c h R / R N v a r i e s f r o m 0.25 to 0.75, w e r e of the o r d e r of t e n t h s of m i l l i d e g r e e s to a few m i l l i - d e g r e e s d e p e n d i n g on R r e s . H e n c e , the f i l m s w e r e of g o o d q u a l i t y and the e f f e c t s of m a c r o s c o p i c i a h o m o g e n e i t i e s w e r e e v i d e n t l y s m a l l . T h e t r a n s i t i o n t e m p e r a t u r e s of all f i l m s w e r e s i m i l a r but i n c r e a s e d s l i g h t l y with Rres. T h e e x p e r i m e n t a l v a l u e of " r o / R r e s was o b t a i n e d by a l e a s t s q u a r e s fit of the d a t a to eq. (1) with T c and R N a s p a r a m e t e r s . It w a s not p o s s i b l e to m e a s u r e R N d i r e c t l y b e c a u s e of the a n n e a l i n g of the f i l m at h i g h e r t e m p e r a t u r e s and the o n s e t of the phonon c o n t r i b u t i o n to R N at about 90°K. In the r e g i o n s w h e r e the fit w a s done s m a l l e r r o r s in R N m a d e l i t t l e d i f f e r e n c e in ~'o" The e x t r a c o n d u c t i v i t y due to f l u c t u a t i o n s , a ' = o"517o/~- , w a s a l s o p l o t t e d a g a i n s t t e m p e r a t u r e and
Volume 28A, n u m b e r 3
PHYSICS
LETTERS
Table 1. d (,~)
ares
are shown in table 1 and agree well with theoretical predictions. T h e d e p a r t u r e f r o m eq. (1) a s r --* 0 i s d u e t o t h e b r e a k d o w n of t h e m e a n f i e l d t h e o r y o n w h i c h the calculations [1,2] are based. For the high c u r r e n t s w h i c h w e h a v e u s e d we m i g h t e x p e c t t h i s to occur when the superfluid velocity exceeds a c r i t i c a l v a l u e g i v e n b y v s ~ ~/(m~) w h e r e ~ i s t h e t e m p e r a t u r e d e p e n d e n t c o h e r e n c e l e n g t h . If w e u s e t h e f o r m u l a t i o n of A b r a h a m s a n d W o o [2], we may estimate the temperature at which this occurs f r o m v S ~ 2eE/(mF) = ~//(m~) w h e r e r-1 = = /f ~ / ( 8 k T r ) i s t h e d e c a y t i m e of a f l u c t u a t i o n [2]. We find
T / a r e s (~'~--1)
(~')
590
3.52
355
14.75
0.155
610
3.46
0.152
416
10.20
0.163
0.158 × 10 -4 ~ 0.004
-
/
/
x J o4J
/
X
x
0
1
10 0
T c ~ [e(~ol)~ E / k B r c ] o -
I
18 November 1968
/
-
x/0 x
2
3
(2)
T h e a r r o w s on fig. 1 i n d i c a t e t h a t t h e v a l u e s of ~ c a g r e e i n t h e i r o r d e r of m a g n i t u d e w i t h eq. (2), a n d t h e E U S f i e l d d e p e n d e n c e of T c i s w e l l r e p r o d u c e d . A l l of t h e f i l m s i n t a b l e 1 e x h i b i t t h i s b e havior.
o o
o
Xo 1o
x~
/ l O- I
,
c°
I
I
I 10 3
I
I
I
(TITc - I )-I
I
I
.
I
i0 d
F i g . 1 . The data shown a r e for a film of t h i c k n e s s 355 A and a r e a = 14.75 ~ , and c u r r e n t d e n s i t i e s J = = 27.2 amp/cm,2 and 4J. The a r r o w s indicate the s t a r t of deviations f r o m theory (dashed line), for the two c u r r e n t densities. the slope found was the same as that obtained f r o m t h e l e a s t s q u a r e s fit. T h i s i n d i c a t e s t h a t t h e r e s u l t s i s n o t a p r o p e r t y of t h e fit. T h e r e s u l t s
References 1. L . G . A s l a m a z o v and A.I. Larkin, Phys. L e t t e r s 26A (1968) 238; H. Sehmidt, to be published. 2. E . A b r a h a m s and J. S . F . Woo, Phys. L e t t e r s 27A (1968) 117. 3. R . E . G l o v e r , Phys. L e t t e r s 25A (1967) 542. 4. M.Strongin, O . F . K a m m e r e r , J . C r o w , R.S. Thompson and H. L. Fine, Phys. Rev. L e t t e r s 20 (1968) 922. 5. J . S . Shier and D. M. Ginsberg, Phys. Rev. 147 (1966) 384.
225
PHYSICS
LETTERS
w h e r e the s y s t e m i s a s s u m e d to h a v e a f i n i t e n u m b e r of a c c e s s i b l e q u a n t u m s t a t e s w e i g h t e d w i t h p r o b a b i l i t i e s P l , 1 = 1 , . . . , n and is o t h e r w i s e uns p e c i f i e d . A n o t h e r b a s i c a s s u m p t i o n r e q u i r e d in t h e u n i q u e n e s s t h e o r e m i s that S { P l } a t t a i n s a m a x i m u m w i t h r e s p e c t to e a c h P l in a n e q u i l i b r i u m s t a t e . T h e q u a n t u m s t a t i s t i c a l e r g o d i c and H t h e o r e m s , b o t h of w h i c h a r e a x i o m a t i c in n a t u r e , a r e l a t e n t i n g r e d i e n t s in the p r o o f of the u n i q u e n e s s of the f o r m S g i v e n a b o v e . O u r a p p r o x i m a t e k i n e t i c t r e a t m e n t is c l e a r l y l e s s g e n e r a l , but
18 N o v e m b e r 1968
d o e s not s e e m to r e l y on t h e s e t h e o r e m s .
References 1. s. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases (Cambridge at the University P r e s s , 1960). 2. L.D. Landau and E. M. Lifshitz, Fluid mechanics (Pergamon P r e s s , 1959) p. 187. 3. P . T . Landsberg, Thermodynamics (Interscience, 1961) Appendix C.
****~
THE
ROUNDING OF THE IN METALLIC
RESISTIVE TRANSITION LEAD FILMS *
R. O. SMITH, B. S E R I N and E. A B R A H A M S Department of Physics, Rutgers University, New Brunswick, NJ. R e c e i v e d 3 S e p t e m b e r 1968
The resistive transition of metallic lead films exhibit the rounding predicted by mean field theories of thermal fluctuations. Close to Tc, the data depend on current density; deviations from the predicted behavior as T--- T c are consistent with the expected breakdown of the theory.
We h a v e i n v e s t i g a t e d the r e s i s t i v e t r a n s i t i o n of m e t a l l i c l e a d f i l m s and h a v e found a r o u n d i n g of the r e s i s t a n c e w h i c h a g r e e s w e l l with t h e o r e t i c a l p r e d i c t i o n s [1,2]. T h e r e s i s t a n c e of a f i l m w i t h t h i c k n e s s d i s p r e d i c t e d to v a r y w i t h t e m p e r a t u r e a c c o r d i n g to RN/R(T)
- 1 = TO/~"
(1)
w h e r e "r = (T - T c ) / T c , TO = R r e s e 2 / ( 1 6 ~ ) and R r e s = 1/(oNd)f~. T h e r a t i o ~ ' o / R r e s is then t h e s a m e f o r a l l f i l m s and e q u a l to 0.152 × 10 - 4 ~ - 1 . W h i l e s o m e p r e v i o u s e x p e r i m e n t a l d a t a [3] on a m o r p h o u s s t r u c t u r e s and v e r y d i r t y s y s t e m s show reasonable agreement with this result, other r e s u l t s [4] a r e u n c l e a r o r d i s a g r e e [5]. H o w e v e r , our results for ~-o/Rres for metallic lead films a r e v e r y c l o s e to t h e t h e o r e t i c a l v a l u e . W e a l s o find that a s ~- ~ 0 d e p a r t u r e s f r o m eq. (1) o c c u r at t e m p e r a t u r e s f o r w h i c h the s u p e r f l u i d v e l o c i t y i s of the o r d e r of a c r i t i c a l v e l o c i t y . * Research supported by the U. S. Army Research and Development Laboratory, Fort Monmounth, N . J . , the National Science Foundation, and the Rutgers Research Council. 224
F i l m s of v a r i o u s t h i c k n e s s e s s w e r e c o n d e n s ed onto a thin g l a s s s u b s t r a t e h e l d at t e m p e r a t u r e s r a n g i n g f r o m 6 to 10OK. T h i s p r o v i d e d r e a s o n a b l e v a r i a t i o n s in the r e s i d u a l r e s i s t a n c e s , R r e s , with w h i c h to t e s t the c a l c u l a t i o n s . T h e c o r r e s p o n d i n g m e a n f r e e p a t h s v a r i e d f r o m 15 to 40 A. T h e t r a n s i t i o n w i d t h s , d e f i n e d by the t e m p e r a t u r e r a n g e in w h i c h R / R N v a r i e s f r o m 0.25 to 0.75, w e r e of the o r d e r of t e n t h s of m i l l i d e g r e e s to a few m i l l i - d e g r e e s d e p e n d i n g on R r e s . H e n c e , the f i l m s w e r e of g o o d q u a l i t y and the e f f e c t s of m a c r o s c o p i c i a h o m o g e n e i t i e s w e r e e v i d e n t l y s m a l l . T h e t r a n s i t i o n t e m p e r a t u r e s of all f i l m s w e r e s i m i l a r but i n c r e a s e d s l i g h t l y with Rres. T h e e x p e r i m e n t a l v a l u e of " r o / R r e s was o b t a i n e d by a l e a s t s q u a r e s fit of the d a t a to eq. (1) with T c and R N a s p a r a m e t e r s . It w a s not p o s s i b l e to m e a s u r e R N d i r e c t l y b e c a u s e of the a n n e a l i n g of the f i l m at h i g h e r t e m p e r a t u r e s and the o n s e t of the phonon c o n t r i b u t i o n to R N at about 90°K. In the r e g i o n s w h e r e the fit w a s done s m a l l e r r o r s in R N m a d e l i t t l e d i f f e r e n c e in ~'o" The e x t r a c o n d u c t i v i t y due to f l u c t u a t i o n s , a ' = o"517o/~- , w a s a l s o p l o t t e d a g a i n s t t e m p e r a t u r e and
Volume 28A, n u m b e r 3
PHYSICS
LETTERS
Table 1. d (,~)
ares
are shown in table 1 and agree well with theoretical predictions. T h e d e p a r t u r e f r o m eq. (1) a s r --* 0 i s d u e t o t h e b r e a k d o w n of t h e m e a n f i e l d t h e o r y o n w h i c h the calculations [1,2] are based. For the high c u r r e n t s w h i c h w e h a v e u s e d we m i g h t e x p e c t t h i s to occur when the superfluid velocity exceeds a c r i t i c a l v a l u e g i v e n b y v s ~ ~/(m~) w h e r e ~ i s t h e t e m p e r a t u r e d e p e n d e n t c o h e r e n c e l e n g t h . If w e u s e t h e f o r m u l a t i o n of A b r a h a m s a n d W o o [2], we may estimate the temperature at which this occurs f r o m v S ~ 2eE/(mF) = ~//(m~) w h e r e r-1 = = /f ~ / ( 8 k T r ) i s t h e d e c a y t i m e of a f l u c t u a t i o n [2]. We find
T / a r e s (~'~--1)
(~')
590
3.52
355
14.75
0.155
610
3.46
0.152
416
10.20
0.163
0.158 × 10 -4 ~ 0.004
-
/
/
x J o4J
/
X
x
0
1
10 0
T c ~ [e(~ol)~ E / k B r c ] o -
I
18 November 1968
/
-
x/0 x
2
3
(2)
T h e a r r o w s on fig. 1 i n d i c a t e t h a t t h e v a l u e s of ~ c a g r e e i n t h e i r o r d e r of m a g n i t u d e w i t h eq. (2), a n d t h e E U S f i e l d d e p e n d e n c e of T c i s w e l l r e p r o d u c e d . A l l of t h e f i l m s i n t a b l e 1 e x h i b i t t h i s b e havior.
o o
o
Xo 1o
x~
/ l O- I
,
c°
I
I
I 10 3
I
I
I
(TITc - I )-I
I
I
.
I
i0 d
F i g . 1 . The data shown a r e for a film of t h i c k n e s s 355 A and a r e a = 14.75 ~ , and c u r r e n t d e n s i t i e s J = = 27.2 amp/cm,2 and 4J. The a r r o w s indicate the s t a r t of deviations f r o m theory (dashed line), for the two c u r r e n t densities. the slope found was the same as that obtained f r o m t h e l e a s t s q u a r e s fit. T h i s i n d i c a t e s t h a t t h e r e s u l t s i s n o t a p r o p e r t y of t h e fit. T h e r e s u l t s
References 1. L . G . A s l a m a z o v and A.I. Larkin, Phys. L e t t e r s 26A (1968) 238; H. Sehmidt, to be published. 2. E . A b r a h a m s and J. S . F . Woo, Phys. L e t t e r s 27A (1968) 117. 3. R . E . G l o v e r , Phys. L e t t e r s 25A (1967) 542. 4. M.Strongin, O . F . K a m m e r e r , J . C r o w , R.S. Thompson and H. L. Fine, Phys. Rev. L e t t e r s 20 (1968) 922. 5. J . S . Shier and D. M. Ginsberg, Phys. Rev. 147 (1966) 384.
225