- Email: [email protected]
Temperature dependence of the magnetic field periodicity in flux quantization experiments
Volume 26A, number 8
TEMPERATURE
PHYSICS
LETTERS
DEPENDENCE OF THE IN FLUX QUANTIZATION
ii March 1968
MAGNETIC FIELD EXPERIMENTS*
PERIODICITY
R. MESERVEY
Francis Bitter National Magnet Laboratory. **, Massachusetts Institute of Technology Cambridge, Massachusetts, USA and L. MEYERS
Boston University, Boston, Massachusetts, USA R e c e i v e d 5 F e b r u a r y 1968
An u n e x p e c t e d t e m p e r a t u r e d e p e n d e n c e of t h e m a g n e t i c field p e r i o d i c i t y h a s b e e n o b s e r v e d in flux q u a n t i z a t i o n e x p e r i m e n t s on s u p e r c o n d u c t i n g t h i n - f i l m hollow c y l i n d e r s of a l u m i n u m . A l t e r n a t e e x p l a n a t i o n s a r e o f f e r e d in t e r m s of v a r i a t i o n s in the t h i c k n e s s o r in t h e o r d e r p a r a m e t e r .
T h e method u s e d in the p r e s e n t e x p e r i m e n t s w a s s i m i l a r to that u s e d by L i t t l e and P a r k s [1] and by M e y e r s and L i t t l e [2]. O u r c y l i n d e r s , which w e r e t y p i c a l l y 1 m m lonog, 0.5 to 5 m i c r o n s in d i a m e t e r , and 400 to 1O00 A in w a l l t h i c k n e s s , w e r e p r e p a r e d b y v a c u u m e v a p o r a t i o n of a l u m i num onto r o t a t i n g q u a r t z f i b e r s . The s a m p l e s w e r e i m m e r s e d in liquid h e l i u m and the t e m p e r a t u r e w a s h e l d fixed (+ 10-5OK) within the r e g i o n w h e r e the l o n g i t u d i n a l r e s i s t a n c e of the c y l i n d e r w a s g r e a t e r than z e r o but l e s s than the n o r m a l r e s i s t a n c e . The r e s i s t a n c e of the s a m p l e w a s p l o t t e d a g a i n s t the a x i a l m a g n e t i c field. F i g u r e 1 shows the x - y recorder traces for a cylinder (diaz~oeter = 0.65 micron, average thickness 500 A) at various different temperatures, and careful examination reveals that there is a small, but definite, increase in the magnetic field period with temperature, the numerical value for this sample being d(AHklt = 32 G/OK. Tinkham [3] has predicted that the transition temperature of such a cylinder in an axially m a g netic field H should change by an amount ATc
=
-
(N~o
-
7rR2H)2/87r2R 2~e2(0)Hcb2(0) • (1)
D o u g l a s s [4] h a s shown that when c o n s i d e r i n g h i g h e r o r d e r e f f e c t s the l o c a l m a x i m a in A Tc ( m i n i m a in r e s i s t a n c e ) should o c c u r a c c u r a t e l y • R e s e a r c h s u p p o r t e d by t h e E l e c t r o n i c s R e s e a r c h C e n t e r u n d e r NASA C o n t r a c t NAS 1 2 - 1 0 1 . • * S u p p o r t e d by t h e U. S. A i r F o r c e Office of S c i e n t i f i c
Research.
OHM
',
I
I
.
I
[ -
[
'
[
:
/-T
I = t,580
--T2 ~.~
~,+
= [. 3 4 9
~. T5 = 1 1 2 8 5
~ T 6 =1.257 T9 =1. 172
~" TIO=~.I40
-400
-200 0 200 MAGNETIC FIELD (GAUSS)
400
Fig. 1. Resistance vers~s magnetic field for a hollow aluminum cylinder 500 A thick, and 0.65 ~ diameter. The upward arrows mark the field difference over 10 periods at various temperatures. at t h o s e f i e l d v a l u e s defined by H m = N~o/zrR a s i m p l i e d b y eq. (1), not withstanding a n o n - p e r i o d i c change in A Tc p r o p o r t i o n a l t o / / 2 . A p o s s i b l e i n t e r p r e t a t i o n of the t e m p e r a t u r e d e p e n d e n c e of the f i e l d p e r i o d i s b a s e d on two w e l l - k n o w n f a c t s . (1) Thin m e t a l f i l m s e v a p o r a t e d onto uncooled s u b s t r a t e s a r e n o n - u n i f o r m in t h i c k n e s s . ( 2 ) F o r a l u m i n u m f i l m s whose t h i c k n e s s d < 1000A, Tc i n c r e a s e s r a p i d l y a s d d e c r e a s e s . Thus the a v e r a g e r a d i u s would be s m a l l e r at h i g h e r t e m p e r a t u r e s and r e s u l t in a l a r g e r p e r i o d . N u m e r i c a l l y t h e o g r e a t e s t change in e f f e c t i v e r a d i u s would b e 85 A. Although the above m o d e l s e e m s p l a u s i b l e , we wish to point out that eq. (1) i s b a s e d on two a s s u m p t i o n s which m a y not be j u s t i f i e d . F i r s t the 367
Volume 26A. n u m b e r 8
PHYSICS
a s s u m e d c o n s t a n c y of t h e o r d e r p a r a m e t e r ( a l though plausible in the radial direction since d << ~(T) * m a y not b e v a l i d i n t h e a z i m u t h a l d i r e c t i o n b e c a u s e of t h i c k n e s s v a r i a t i o n s . S u c h v a r i a t i o n s in I,~f2 w o u l d p r o b a b l y b e g r e a t e s t a t t h e lower temperatures and lead to increased phase change around the cylinder resulting in a relativ e l y h i g h e r f i e l d p e r i o d at h i g h e r t e m p e r a t u r e s , as observed. Secondly the assumed time indep e n d e n c e of t h e o r d e r p a r a m e t e r m a y n o t b e j u s t i f i e d i n t h e t h i n n e s t r e g i o n s n e a r Tc b e c a u s e of fluctuations. Presumably such fluctuations could also affect phase change around the cylinder. • A t e m p e r a t u r e dependence of the filed periodicity in r e c t a n g u l a r loops was previopslv o b s e r v e d by one of us [5] but in that case d~60O0 ./k and a v a r i a t i o n in I~t 2 p e r p e n d i c u l a r to the film plane s e e m e d very plausible.
X-POINT
SHIFT
IN
LETTERS
11 March 1968
W e c o n c l u d e t h a t , a l t h o u g h t h e c a u s e of t h i s temperature variation needs further investigat i o n , i t s e x i s t e n c e s h o u l d b e c o n s i d e r e d in p r e c i s i o n m e a s u r e m e n t s of hc/2e b a s e d o n t h e m a g n e t i c f i e l d p e r i o d i c i t y in t h i n f i l m s u p e r c o n d u c t ing cylinders. 1. W.A. Little and R. D. P a r k s , Phys. Rev. L e t t e r s 9 (1962) 9: R. D. P a r k s and W. A. Little, Phys. Re:v. 133 (1964) A97. 2. L. M e y e r s and W. A. Little, Phys. Rev. J~etters 13 (1964) 325. 3. M. Tinkham, Phys. Rev. 129 (1963) 2413. 4. D.H. Douglass J r . . Phys. Rev. 132 (1963) 513. 5. R. Meservey, Proc. 9th Intern. Conf. on Low temp. physics, Columbus (1964) (Plenum P r e s s , P a r t A) p.455.
TURBULENTLY
FLOWING
4He
K. D. E R B E N a n d F. P O B E L L
Physik Dept. der Technischen Hochschule Miinchen, Germany Received 12 F e b r u a r y 1968
Shifts of the k - t e m p e r a t u r e of 4He up to A T k = 1.4 × 1 0 - 3 ° K due to turbulent flow caused by a heat c u r r e n t in a wide tube have been observed.
Recently there has been considerable interest i n d e t e c t i n g a s h i f t of t h e X - p o i n t due to t h e c r e a t i o n of q u a n t i z e d v o r t i c e s in r o t a t i n g l i q u i d 4He [1,2]. T h e k - p o i n t w a s f o u n d t o b e i n d e p e n d e n t of r o t a t i o n w i t h i n 10 - 5 OK u p t o a n a n g u l a r v e l o c i t y of co = 110 s e c -1 [1,2]. T h e p h e n o m e n o l o g i c a l t h e o r y [3,4] p r e d i c t s t h a t t h e s e a n g u l a r v e l o c i t i e s a r e t o o s m a l l to p r o d u c e m e a s u r a b l e s h i f t s of Tk [5,6]. H o w e v e r , a h i g h e r a v e r a g e v e l o c i t y of t h e s u p e r f l u i d c o m p o n e n t can b e a t t a i n e d , p a r t i c u larly just below the k-point, by a heat current. H e r e w e r e p o r t o n a s h i f t of t h e k - p o i n t u p to
o
A T x = 1.4 × 10 -3 O K c a u s e d by a heat current in
z o
helium contained in a very wide tube. The order of m a g n i t u d e of t h i s s h i f t i s i n a g r e e m e n t w i t h t h e o r e t i c a l e x p e c t a t i o n s [5,6]. Fig. 1 shows the bucket which is immersed into the helium bath. Helium was introduced into c h a m b e r A (1.40 c m c m i. d . , 9.6 c m l o n g ) t h r o u g h valve V until the meniscus could be seen in the g l a s s c a p i l l a r y C. D u e t o t h e v a c u u m b e t w e e n chambers A and B, thermal contact between the 368
2
~---
~_ A
C
/.-
R
J
e T ~ e e •. •f
LD
Q
ee i
20 ]
HEATING
30 I
RATE
40 I
[mW
50 J
60 I
cm -2 ]
Fig. 1. Shift of the k-point t e m p e r a t u r e of 4He as a function of the heat input at the bottom of c h a m b e r A. Insert: Experimental set-up.
TEMPERATURE
PHYSICS
LETTERS
DEPENDENCE OF THE IN FLUX QUANTIZATION
ii March 1968
MAGNETIC FIELD EXPERIMENTS*
PERIODICITY
R. MESERVEY
Francis Bitter National Magnet Laboratory. **, Massachusetts Institute of Technology Cambridge, Massachusetts, USA and L. MEYERS
Boston University, Boston, Massachusetts, USA R e c e i v e d 5 F e b r u a r y 1968
An u n e x p e c t e d t e m p e r a t u r e d e p e n d e n c e of t h e m a g n e t i c field p e r i o d i c i t y h a s b e e n o b s e r v e d in flux q u a n t i z a t i o n e x p e r i m e n t s on s u p e r c o n d u c t i n g t h i n - f i l m hollow c y l i n d e r s of a l u m i n u m . A l t e r n a t e e x p l a n a t i o n s a r e o f f e r e d in t e r m s of v a r i a t i o n s in the t h i c k n e s s o r in t h e o r d e r p a r a m e t e r .
T h e method u s e d in the p r e s e n t e x p e r i m e n t s w a s s i m i l a r to that u s e d by L i t t l e and P a r k s [1] and by M e y e r s and L i t t l e [2]. O u r c y l i n d e r s , which w e r e t y p i c a l l y 1 m m lonog, 0.5 to 5 m i c r o n s in d i a m e t e r , and 400 to 1O00 A in w a l l t h i c k n e s s , w e r e p r e p a r e d b y v a c u u m e v a p o r a t i o n of a l u m i num onto r o t a t i n g q u a r t z f i b e r s . The s a m p l e s w e r e i m m e r s e d in liquid h e l i u m and the t e m p e r a t u r e w a s h e l d fixed (+ 10-5OK) within the r e g i o n w h e r e the l o n g i t u d i n a l r e s i s t a n c e of the c y l i n d e r w a s g r e a t e r than z e r o but l e s s than the n o r m a l r e s i s t a n c e . The r e s i s t a n c e of the s a m p l e w a s p l o t t e d a g a i n s t the a x i a l m a g n e t i c field. F i g u r e 1 shows the x - y recorder traces for a cylinder (diaz~oeter = 0.65 micron, average thickness 500 A) at various different temperatures, and careful examination reveals that there is a small, but definite, increase in the magnetic field period with temperature, the numerical value for this sample being d(AHklt = 32 G/OK. Tinkham [3] has predicted that the transition temperature of such a cylinder in an axially m a g netic field H should change by an amount ATc
=
-
(N~o
-
7rR2H)2/87r2R 2~e2(0)Hcb2(0) • (1)
D o u g l a s s [4] h a s shown that when c o n s i d e r i n g h i g h e r o r d e r e f f e c t s the l o c a l m a x i m a in A Tc ( m i n i m a in r e s i s t a n c e ) should o c c u r a c c u r a t e l y • R e s e a r c h s u p p o r t e d by t h e E l e c t r o n i c s R e s e a r c h C e n t e r u n d e r NASA C o n t r a c t NAS 1 2 - 1 0 1 . • * S u p p o r t e d by t h e U. S. A i r F o r c e Office of S c i e n t i f i c
Research.
OHM
',
I
I
.
I
[ -
[
'
[
:
/-T
I = t,580
--T2 ~.~
~,+
= [. 3 4 9
~. T5 = 1 1 2 8 5
~ T 6 =1.257 T9 =1. 172
~" TIO=~.I40
-400
-200 0 200 MAGNETIC FIELD (GAUSS)
400
Fig. 1. Resistance vers~s magnetic field for a hollow aluminum cylinder 500 A thick, and 0.65 ~ diameter. The upward arrows mark the field difference over 10 periods at various temperatures. at t h o s e f i e l d v a l u e s defined by H m = N~o/zrR a s i m p l i e d b y eq. (1), not withstanding a n o n - p e r i o d i c change in A Tc p r o p o r t i o n a l t o / / 2 . A p o s s i b l e i n t e r p r e t a t i o n of the t e m p e r a t u r e d e p e n d e n c e of the f i e l d p e r i o d i s b a s e d on two w e l l - k n o w n f a c t s . (1) Thin m e t a l f i l m s e v a p o r a t e d onto uncooled s u b s t r a t e s a r e n o n - u n i f o r m in t h i c k n e s s . ( 2 ) F o r a l u m i n u m f i l m s whose t h i c k n e s s d < 1000A, Tc i n c r e a s e s r a p i d l y a s d d e c r e a s e s . Thus the a v e r a g e r a d i u s would be s m a l l e r at h i g h e r t e m p e r a t u r e s and r e s u l t in a l a r g e r p e r i o d . N u m e r i c a l l y t h e o g r e a t e s t change in e f f e c t i v e r a d i u s would b e 85 A. Although the above m o d e l s e e m s p l a u s i b l e , we wish to point out that eq. (1) i s b a s e d on two a s s u m p t i o n s which m a y not be j u s t i f i e d . F i r s t the 367
Volume 26A. n u m b e r 8
PHYSICS
a s s u m e d c o n s t a n c y of t h e o r d e r p a r a m e t e r ( a l though plausible in the radial direction since d << ~(T) * m a y not b e v a l i d i n t h e a z i m u t h a l d i r e c t i o n b e c a u s e of t h i c k n e s s v a r i a t i o n s . S u c h v a r i a t i o n s in I,~f2 w o u l d p r o b a b l y b e g r e a t e s t a t t h e lower temperatures and lead to increased phase change around the cylinder resulting in a relativ e l y h i g h e r f i e l d p e r i o d at h i g h e r t e m p e r a t u r e s , as observed. Secondly the assumed time indep e n d e n c e of t h e o r d e r p a r a m e t e r m a y n o t b e j u s t i f i e d i n t h e t h i n n e s t r e g i o n s n e a r Tc b e c a u s e of fluctuations. Presumably such fluctuations could also affect phase change around the cylinder. • A t e m p e r a t u r e dependence of the filed periodicity in r e c t a n g u l a r loops was previopslv o b s e r v e d by one of us [5] but in that case d~60O0 ./k and a v a r i a t i o n in I~t 2 p e r p e n d i c u l a r to the film plane s e e m e d very plausible.
X-POINT
SHIFT
IN
LETTERS
11 March 1968
W e c o n c l u d e t h a t , a l t h o u g h t h e c a u s e of t h i s temperature variation needs further investigat i o n , i t s e x i s t e n c e s h o u l d b e c o n s i d e r e d in p r e c i s i o n m e a s u r e m e n t s of hc/2e b a s e d o n t h e m a g n e t i c f i e l d p e r i o d i c i t y in t h i n f i l m s u p e r c o n d u c t ing cylinders. 1. W.A. Little and R. D. P a r k s , Phys. Rev. L e t t e r s 9 (1962) 9: R. D. P a r k s and W. A. Little, Phys. Re:v. 133 (1964) A97. 2. L. M e y e r s and W. A. Little, Phys. Rev. J~etters 13 (1964) 325. 3. M. Tinkham, Phys. Rev. 129 (1963) 2413. 4. D.H. Douglass J r . . Phys. Rev. 132 (1963) 513. 5. R. Meservey, Proc. 9th Intern. Conf. on Low temp. physics, Columbus (1964) (Plenum P r e s s , P a r t A) p.455.
TURBULENTLY
FLOWING
4He
K. D. E R B E N a n d F. P O B E L L
Physik Dept. der Technischen Hochschule Miinchen, Germany Received 12 F e b r u a r y 1968
Shifts of the k - t e m p e r a t u r e of 4He up to A T k = 1.4 × 1 0 - 3 ° K due to turbulent flow caused by a heat c u r r e n t in a wide tube have been observed.
Recently there has been considerable interest i n d e t e c t i n g a s h i f t of t h e X - p o i n t due to t h e c r e a t i o n of q u a n t i z e d v o r t i c e s in r o t a t i n g l i q u i d 4He [1,2]. T h e k - p o i n t w a s f o u n d t o b e i n d e p e n d e n t of r o t a t i o n w i t h i n 10 - 5 OK u p t o a n a n g u l a r v e l o c i t y of co = 110 s e c -1 [1,2]. T h e p h e n o m e n o l o g i c a l t h e o r y [3,4] p r e d i c t s t h a t t h e s e a n g u l a r v e l o c i t i e s a r e t o o s m a l l to p r o d u c e m e a s u r a b l e s h i f t s of Tk [5,6]. H o w e v e r , a h i g h e r a v e r a g e v e l o c i t y of t h e s u p e r f l u i d c o m p o n e n t can b e a t t a i n e d , p a r t i c u larly just below the k-point, by a heat current. H e r e w e r e p o r t o n a s h i f t of t h e k - p o i n t u p to
o
A T x = 1.4 × 10 -3 O K c a u s e d by a heat current in
z o
helium contained in a very wide tube. The order of m a g n i t u d e of t h i s s h i f t i s i n a g r e e m e n t w i t h t h e o r e t i c a l e x p e c t a t i o n s [5,6]. Fig. 1 shows the bucket which is immersed into the helium bath. Helium was introduced into c h a m b e r A (1.40 c m c m i. d . , 9.6 c m l o n g ) t h r o u g h valve V until the meniscus could be seen in the g l a s s c a p i l l a r y C. D u e t o t h e v a c u u m b e t w e e n chambers A and B, thermal contact between the 368
2
~---
~_ A
C
/.-
R
J
e T ~ e e •. •f
LD
Q
ee i
20 ]
HEATING
30 I
RATE
40 I
[mW
50 J
60 I
cm -2 ]
Fig. 1. Shift of the k-point t e m p e r a t u r e of 4He as a function of the heat input at the bottom of c h a m b e r A. Insert: Experimental set-up.