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Shape resonances in superconductors - II simplified theory
Volume 5, number 1
PHYSICS
e f f e c t s m i g h t p o s s i b l y a r i s e f r o m f i l l f l o w of t h e He 4 r i c h p h a s e , we have r e p e a t e d the e x p e r i m e n t s
w i t h e x t r a high p u r i t y He 3 , a n d w i t h He 3 c o n t a i n i n g 0.07% He 4 in w h i c h one w o u l d e x p e c t t h e p h a s e s e p a r a t i o n to o c c u r a t a b o u t 0.15OK 3). A l l t h e s e e x p e r i m e n t s g a v e s i m i l a r r e s u l t s , a s shown in fig. 2. L a n d a u ' s t h e o r y of a F e r m i l i q u i d 4) l e a d s to t h e c o n c e p t of a r e l a x a t i o n t i m e T c h a r a c t e r i s t i c of t h e c o l l i s i o n s b e t w e e n He 3 a t o m s (or m o r e a c c u r a t e l y b e t w e e n q u a s i - p a r t i c l e s ) w h i c h give r i s e to t h e v i s c o s i t y . A n a l y s i s of t h e v i s c o s i t y and o t h e r t r a n s p o r t p r o p e r t i e s s h o w s t h a t T i n c r e a s e s r a p i d l y with f a l l i n g t e m p e r a t u r e and a t 0 . 0 9 ° K h a s a v a l u e s u c h t h a t o~T ~ 1, w h e r e o~ i s the a n g u l a r f r e q u e n c y of t h e 1000 M c / s sound w a v e . H e n c e t h e a b r u p t c h a n g e in t h e v a l u e of Z/p o c c u r s when t h e m e a n f r e e p a t h b e t w e e n collisions becomes comparable with the wave length of t h e sound. T h e o n l y t r e a t m e n t of t h e p r o p a g a t i o n of sound in He 3 a t v e r y low t e m p e r a t u r e s i s t h a t of L a n d a u 5) b a s e d on h i s t h e o r y of a F e r m i l i q u i d . B e c a u s e of t h e long r e l a x a t i o n t i m e b e t w e e n c o l l i s i o n s , a sound w a v e c a n n o t p r o p a g a t e in the n o r m a l w a y , b u t L a n d a u p r e d i c t s t h a t t h e r e w i l l be a new c o l l e c t i v e m o d e of p r o p a g a t i o n . T h i s new m o d e (which he t e r m s z e r o sound) i s c h a r a c t e r i s e d b y a d i f f e r e n t v e l o c i t y a n d a d i f f e r e n t c o e f f i c i e n t of a b s o r p t i o n , but t h e s e q u a n t i t i e s a r e v e r y d i f f i c u l t to m e a s u r e on a c c o u n t of t h e high f r e q u e n c i e s and low t e m p e r a t u r e i n v o l v e d . H o w e v e r , i t a p p e a r s t h a t in the present experiment we are observing the trans i t i o n f r o m o r d i n a r y to z e r o sound. In the r e g i o n of z e r o sound t h e c l a s s i c a l r e l a t i o n b e t w e e n a c o u s t i c i m p e d a n c e and v e l o c i t y of sound ( Z = pc) i s no l o n g e r v a l i d . T h e r e f o r e w e c a n n o t
SHAPE
RESONANCES IN SIMPLIFIED
LETTERS
1 June 1963
i n t e r p r e t t h e v a l u e s of Z/p b e l o w the t r a n s i t i o n d i r e c t l y a s the v e l o c i t y of z e r o sound (Co). E x a c t a l g e b r a i c e x p r e s s i o n s g i v i n g the r e l a t i o n b e t w e e n Z / p and co h a v e b e e n g i v e n b y B e k a r a v i c h and K h a l a t n i k o v 6) in c o n n e c t i o n w i t h t h e i r c a l c u l a t i o n of t h e t h e r m a l b o u n d a r y r e s i s t a n c e b e t w e e n s o l i d s a n d l i q u i d He 3. A t p r e s e n t M r . G. A. B r o o k e r of t h i s l a b o r a t o r y i s e n g a g e d in t h e l e n g t h y p r o c e s s of o b taining numerical values for these expressions, using the latest values for.the various parameters c h a r a c t e r i s i n g t h e l i q u i d 7). So f a r , i t a p p e a r s t h a t f o r e x c i t a t i o n s in the liquid w h i c h a r e s p e c u l a r l y r e f l e c t e d a t the b o u n d a r y , Z/O i s of the o r d e r of c o. H o w e v e r , w h e t h e r Zip i s l e s s t h a n o r g r e a t e r t h a n c o a p p e a r s to d e p e n d r a t h e r c r i t i c a l l y on the v a l u e s of t h e a b o v e m e n t i o n e d p a r a m e t e r s . F u l l d e t a i l s of all this work will be published subsequently. W e a r e m u c h i n d e b t e d to t h e D . S . I . R . f o r a g r a n t f o r e q u i p m e n t , f o r t h e s u p p o r t of one of u s ( P W M ) , and f o r a r e s e a r c h s t u d e n t s h i p (BEK).
References 1) E.H.Jacobsen, Quantum electronics, ed. Townes (Columbia University P r e s s , New York, 1960) p. 468. 2) B. E. Keen, P.W. Matthews and J.Wilks, l>roc, of the VIIIth Int. Conf. on Low Temperature Physics, In p r e s s . 3) D.O. Edwards and J. G. Daunt, Phys. Rev. 124 (1961) 640. 4) L.D. Landau, J. Exptl. Theoret. Phys. (USSR) 30 (1956) 1058; translation: Soviet Phys. JETP 3 (1957) 920. 5) L.D. Landau, J. Exptl. Theoret. Phys. (USSR) 32 (1957) 59; translation: Soviet Phys. JETP 5 (1957) 101. 6) I. L. Bekaravich and I. M. Khalatnikov, J. Exptl. Theoret, P h y s . 39 (1960) 1699; translation: Soviet Phys. JETP 12 (1961) 1187. 7) D.Hone, Phys. Rev. 125 (1962) 1494.
SUPERCONDUCTORS THEORY *
-
II
C. J. T H O M P S O N a n d J. M. B L A T T
Applied Mathematics Department, University of New South Wales, Kensington, Australia Received 23 April 1963
In an e a r l i e r c o m m u n i c a t i o n 1) we r e p o r t e d t h e r e s u l t s of a c o m p u t e r c a l c u l a t i o n f o r a s u p e r c o n ducting uniform membrane, showing pronounced r e s o n a n c e s of t h e e n e r g y gap a s a f u n c t i o n of t h e width a of t h e m e m b r a n e . In o r d e r to g a i n a b e t t e r * This r e s e a r c h has been supported in part by U. S. A i r Force Research Grant No. 62-400 to the University of New South W a l e s .
i n s i g h t into the r e l e v a n t f a c t o r s , we h a v e p e r f o r m e d an a n a l y t i c c a l c u l a t i o n u n d e r c e r t a i n , r e a s o n a b l e , s i m p l i f y i n g a s s u m p t i o n s . In t h i s l e t t e r we p r e s e n t t h i s w o r k . If t h e s l a b e x t e n d s in t h e x - d i r e c t i o n f r o m x = 0 to x = a , and if w e i m p o s e p e r i o d i c b o u n d a r y c o n d i t i o n s in the y - and z - d i r e c t i o n s , with p e r i o d i c i t y d i s t a n c e L, t h e b a s i c o n e - e l e c t r o n w a v e f u n c t i o n
V o l u m e 5, n u m b e r 1
PHYSICS
LETTERS
1 J u n e 1963
has the f o r m (spin functions omitted):
a
Ck(r) = un(x ) exp (ik2Y + i k 3 z ) l L ,
C n = K -1~_, C n , A n , f
it)
where k 2 and k3 a r e integral multiples of 2 y / L and, for the sake of simplicity, un(0 ) = un(a ) = 0. Except for unimportant details, the boundary conditions used do not affect the final results. The functions ~k m a y be taken to be solutions of a Schr~Minger equation -(~2/2M) V2~ok + V ( x ~ k = CkCPk ,
(2)
nt
where (k F = F e r m i momentum, p = usual nondimensional coupling p a r a m e t e r = J P E / V )
ck = r/n + (ti2/2ND(k 2 + k~) ,
and A n = a r c sinh ( k ~ c / C n )
for
=0
tion of an electron p a i r . A n d e r s o n ' s 2) suggestion that the pairing should involve time r e v e r s e d functions (n, k2, k3, l) and (n, -k2, -k3, ~) is c l e a r l y app r o p r i a t e here f r o m s y m m e t r y . The choice of a model interaction and associated reduced m a t r i x elements is somewhat m o r e delicate since a simple delta-function potential
v ( r i - rj) = - gS(r i - rj) ,
(5)
leads to divergent integrals in superconductivity theory. We follow Gor'kov 3) by imposing an e n e r gy-dependent cut-off on the m a t r i x elements, obtained by s t r a i g h t f o r w a r d calculation from (1) and (5). Letting/~ be the chemical potential, the m a t r i x elements in question a r e
N = 2 ~k hk , where h k is related to Ck-/~ and Ck in the usual fashion, 3,4). By p e r f o r m i n g the sums over k 2 and k 3 analytically, we obtain
~ Hn ,
(12)
1"l
I Ck,-~l < tZ~c,
otherwise.
Cn (independent of k2, k3) for I ck-~] < otherwise .
(II)
In p r a c t i c e , we a r e given the number density N / V , not the chemical potential ~. The relation between the two quantities is given by
N/V = (M/2yti2a) Hn =-2(rbw~)
= ~
~c, (7)
This r e p r e s e n t s a highly "anisotropic gap". In the usual integral equation 3,4) of superconductivity theory, one can p e r f o r m the sums over k 2 and k 3 analytically. The r e s u l t is
for
~n-# < - ~ c
,
- [ c.2 + (~c)21 ~ - (~.-.) + [ c~ + (~ _.)2]½ for I~.-~1 < ~c,
(6)
Although the integral depends on n and n' only, the m a t r i x elements t h e m s e l v e s depend also on k2, k3, k~ and k~ because of the cut-off condition. In the limit of large a, f o r m u l a (6) yields - J / L 2 a = - J / V within the allowed region for k ¢ k' (the only inter e s t i n g case). This is the model interaction used by Bardeen et al. 4). With m a t r i x elements of the f o r m (6), the energy gap function Ck b e c o m e s
=0
#v < ~ + ~ c < #v+l •
where
and
=
rb~-/~ > ~ v .
Although the sum over n' in (8) extends over all eigenvalues of eq. (4), we see f r o m (10) that it is actually a finite sum. The maximum value of n' contributing to (8) is the integer v defined by the condition
[Un(X)Un,(X)] 2 dx if both I ck - u l < PiCOc
=0
,
(10)
G o
Ck
for
(4)
In the superconducting state, the one-electron functions (I) a s s o c i a t e in p a i r s to f o r m a wave func-
- ( J / L 2) f
r/n-/~ < - ~ c ,
for It/n-/.ll < ~ c
(3)
- (~2/2M)(d2un/dx 2) + V(x)un(x) = ~nUn(X).
=
(9)
K = 2~tE2/JM = k F / ~ p
= -a ' a r c sinh ( ~ c / C n ) - ½ a r c sinh [(nn -/~)1 Cn ]
which r e d u c e s , together with (1), to
(8)
[un(x)un,(x)] 2{ix ,
0
=0
for
~?n-/~ > ~u~c .
(13) So f a r , our f o r m u l a e have been quite general. In the computer calculation, the potential V(x) in the SchrSdinger equation (4) was taken to be the e l e c t r o static energy of one electron in the e l e c t r o s t a t i c potential produced by the c h a r g e density of all the other electrons, plus a uniformly s m e a r e d out background positive charge. (4) was then solved, the m a t r i x elements were found f r o m (6), and eqs. (8) through (13) were solved with these m a t r i x elements. The computer then determined the new charge distribution, and iterated until a s e l f - c o n sistent solution was achieved. It was found, not s u r p r i s i n g l y , that the ultimate s e l f - c o n s i s t e n t solutions had nearly vanishing
Volume 5, number 1
PHYSICS
LETTERS
I
c h a r g e d e n s i t y t h r o u g h o u t t h e i n t e r i o r of t h e s l a b . T h e b o u n d a r y r e g i o n , in w h i c h C o u l o m b e f f e c t s a r e i m p o r t a n t , w a s found to h a v e a w i d t h "of a p p r o x i m a t e l y 2 Angstr{~m u n i t s , f o r r e a s o n a b l e p a r a m e t e r s . It i s t h e r e f o r e r e a s o n a b l e to ignore the potential
-5~0o[ k
V(x) altogether in order to gain a better qualitative insight. T h i s i s t h e b a s i s of t h e p r e s e n t c a l c u l a t i o n . W i t h V(x) = O, eq. (41 h a s t h e i m m e d i a t e s o l u t i o n Un(X I = (2/a)½ sin (n~rx/a) ,
(14) n=I,2,3~...
r~
---(h2/2M)(n'~/a) 2
1 June. 1963
,
,
(15)
which leads to the overlap integrals a
f
[un(X)Un,(X)] 2 dx = ( l / a ) ( 1 + ½ 5nn,).
(16) THICKNESS OF SLAB ( A U ) - -
o
Fig. 1. Chemical potential tL versus thickness of film. The cusps occur at the lower ends of the resonance r e gions, a is continuous a c r o s s the resonance r e gions and approaches the F e r m i energy eF for large thicknesses. The p a r a m e t e r used for this figure was: N/V = 2 x 1022 e l e c t r o n s / c m 3.
S u b s t i t u t i o n of (16) into (8) y i e l d s [1 - (An/2Ka) ] Cn = (1/Ka) ~, Cn, A n, n'=l = i n d e p e n d e n t of n,
(17)
w h e r e t h e u p p e r l i m i t v i s now g i v e n b y s u b s t i t u t i o n of ( 1 5 ) i n t o (ii):
t
1
v = integral part of {~z/~)[(2M/~2)(~+ ~ c ) ] ~ } .
(18)
A look at (10) s h o w s that A n is a function of C n only, except w h e n ]r~n-~] < ~c~c. F o r m o s t slab widths a, this latter case does not occur for any value of n (the n a r r o w regions of a in which this case does occur correspond to the exceedingly sharp resonance rises in Cn). If we r e s t r i c t our-
w
selves to the "ordinary" regions (no I??n-~[ s m a l l e r t h a n ~O~c), eq. (17) i m p l i e s t h a t Cn i s i n d e p e n d e n t of n , with the v a l u e i0
Cn = ~U~c/sinh [ga/(v+½)] =0
f o r n ..< v , n>/~
6a 3
(201
T h e p r a c t i c a l e v a l u a t i o n of e q s . (18), (191 and (20) s t a r t s f r o m a c h o i c e of t h e i n t e g e r v. By i g n o r i n g t h e s m a l l q u a n t i t y ~coc in (18), we c a n e l i m i n a t e ~ f r o m (18) and (20) t o g e t a l o w e r bound on t h e s l a b w i d t h a f o r t h i s v a l u e of u. If t h i s l o w e r bound i s d e n o t e d b y a p , a~ i s g i v e n b y = -~-~- ~p 3 - ~ '
Z~
(19)
.
v(v+½)(v+11} •
2O
THLCKNESS OF ~LAB ~,A.U)~
Eq. (12) c a n b e s o l v e d f o r # e x p l i c i t l y w i t h the r e sult
U = (~api2/uM) {--N V+~
i~
(21)
W i t h t h e c h o s e n v a l u e of v ~ - f o r m u l a (191 i s a p p l i c a b l e f o r s l a b w i d t h s in t h e r a n g e av+c(a) < a < au+ 1-c(a), w h e r e c(al i s a m e a s u r e of t h e s i z e of t h e r e s o n a n c e r e g i o n s . F o r e a c h v a l u e of a in the
Fig. 2. Superconducting energy gap p a r a m e t e r versus thickness of film. At each resonance a new value of n s t a r t s to contribute. The new values of n are shown on the figure. The peak heights lie well. above the bulk value C~. The troughs a r e only slightly below C=. The widths of the resonances range from ~ 0.04 A at the v = 3 resonance, to 0.1 A at the v = 8 resonance. The parameters used for this figure were: N / V = 2 x 1022 electrons/cm 3, p = 0.3, ~w c = 100OK.
r e l e v a n t r a n g e , we find the c h e m i c a l p o t e n t i a l f r o m (20), and t h e e n e r g y gap p a r a m e t e r f r o m (191. T h e r e s u l t s a r e shown in f i g s . 1 and 2 f o r t h e s p e c i a l case, ~oJc = 100OK, N / V = 2 × 1022 c m - 3 , and 0 = 0.3. F o r these values, ((a) ~ 0.02 a.u. for a = 8 a.u., and ((a I ~ 0.05 a.u. for a = 30 a.u. 11. T h e similarity of fig. 2 to the r e s u R s of the c o m puter calculation is striking. W e note f r o m fig. 1 that there is a secular rise in ~ as the slab is m a d e narrower. This is not a
Volume 5, number 1
PHYSICS
s p e c i a l e f f e c t in s u p e r c o n d u c t i v i t y , b u t o c c u r s a l s o in t h e n o r m a l s t a t e ( F e r m i s e a ) . T h e e n e r g y n e e d e d to p u s h one m o r e p a r t i c l e into t h e v o l u m e V = aL2 i s l a r g e r f o r a c o m p a r a b l e to a F e r m i w a v e l e n g t h , t h a n f o r a = L = VL T h e p e c u l i a r b e h a v i o u r of ~ a s a f u n c t i o n of a, s h o w n in f i g . 1, i s , h o w e v e r , e s s e n t i a l to o b t a i n t h e d e c i d e d a s y m m e t r y of t h e r e s o n a n c e s in f i g . 2. (If w e a s s u m e , i n c o r r e c t l y , t h a t / ~ equals the conventional Fermi energy ~2kF2/2M, t h e r e s o n a n c e p e a k s and t r o u g h s a r e found to b e a l m o s t s y m m e t r i c a l a r o u n d the b u l k v a l u e Coo of t h e e n e r g y gap). The narrow resonance regions can also be disc u s s e d b y a m o r e d e t a i l e d s o l u t i o n of (17) in t h e r e l e v .ant r e g i o n s . No new q u a l i t a t i v e p o i n t s e m e r g e . F i n a l l y , w e e x a m i n e t h e a s y m p t o t i c b e h a v i o u r of the e n e r g y gap and t h e width of t h e r e s o n a n c e r e g i o n s f o r l a r g e s l a b w i d t h s a ( i . e . , l a r g e v). F o r l a r g e v, (21) y i e l d s av ~ v ( # V I 3 N ) { = v ( ~ / k F ) .
(22)
T h e r e s o n a n c e s , t h e r e f o r e , o c c u r a t s l a b w i d t h s a, ectnal to an i n t e g r a l n u m b e r of half w a v e l e n g t h s ~ / k F of a n e l e c t r o n r i g h t a t the F e r m i s u r f a c e . Eq. (20) y i e l d s / ~ ~ ~ 2 k F 2 / 2 M = cF. T h e b u l k v a l u e of t h e e n e r g y gap i s coo
-
~-~c sinh (l/p) "
(23)
It is interesting to observe that the ratio Cn/Coo is independent of the cutoff energy ~.0c whenever (10) is valid, i.e., for all values of the thickness a other than the narrow regions corresponding to the resonance rises in C n. The strength pararneter p which does enter the ratio Cn/Coo measures the strength of the interaction right at the F e r m i surface. If w e define "peak" and "trough" values of the energy gap at resonance n u m b e r v through (cf.(19)) kFa v = I~/.~c]sL~ { ~ } ,
THE
CRITICAL
CURRENT
(24a)
OF
LETTERS
1June 1963
and
C T ,v = ~ c/sinh { _ ~ } ,
(24b)
respectively, and m a k e use of (22), the asymptotic forms of Cp,v/C~o and CT,v/Coo for large v are given, in the weak coupling limit p << 1, by
cp ,,,I
1
e..-'p
(2s)
and
1}
CT,v/Coo~e x p { -
p(2~-l)
(26)
"
T h e s e go to u n i t y f o r l a r g e s l a b t h i c k n e s s , t h e e x c e s s b e i n g i n v e r s e l y p r o p o r t i o n a l to t h e t h i c k n e s s . To o b t a i n t h e a s y m p t o t i c f o r m of t h e r e s o n a n c e w i d t h f o r l a r g e v, w e f i r s t o b s e r v e t h a t t h e u p p e r and l o w e r v a l u e s of a , a U a n d a L , a t r e s o n a n c e n u m b e r v, are defined by ~ - # = - ~o~c and ~v - /~ = + ~/COcrespectively. If w e use these facts together with (15) and the asymptotic value CF for #, w e get aU- a L~Pv
,
(2'/)
where B
~1
.V. 2M
F o r t h e p a r a m e t e r s u s e d in fig. 2, B ~ 1.2 x 10 . 2 A. Unlike t h e r a t i o Cn/Coo, a U - a L i s c u t - o f f d e p e n d e n t (and i n d e p e n d e n t of p). In f a c t , (2'/) s h o w s t h a t for large slab thickness, a U - a L is directly prop o r t i o n a l to ~ c . References 1) J . M . Blatt and C.J.Thompson, Phys. Rev. Letters 10 (1963) 332. 2) P.W.Anderson, J. Phys. Chem. Solids 11 (1959) 26. 3) L. P. Gor'kov, J. Exptl. Theoret. Phys. (USSR) 34 (1958) 735; translation: Soviet Phys. JETP 7 (1958) 505. 4) J. Bardeen, L.N. Cooper and J. R. Schrieffer, Phys. Rev. 108 (1957) 1175.
A SUPERCONDUCTOR
OF
THE
SECOND
KIND
R. A. K A M P E R C. E. R. L., Leatherhead Received 23 April 1963 T h e p u r p o s e of t h i s n o t e i s to r e p o r t an e x p e r i m e n t a l d e m o n s t r a t i o n t h a t a s u p e r c o n d u c t o r with a negative interphase surface energy cannot carry a n e t c u r r e n t in the p r e s e n c e of an e x t e r n a l m a g n e t i c f i e l d s u f f i c i e n t l y s t r o n g to p e n e t r a t e t h e i n t e r i o r
and c r e a t e t h e " m i x e d s t a t e " 1) w i t h o u t t h e a i d of s t r u c t u r a l i m p e r f e c t i o n s to s t a b i l i s e t h e c u r r e n t a g a i n s t t h e L o r e n t z f o r c e 2). If t h i s a s s e r t i o n i s t r u e then t h e c r i t i c a l c u r r e n t I c of a c y l i n d r i c a l s p e c i m e n of r a d i u s r in a t r a n s -
PHYSICS
e f f e c t s m i g h t p o s s i b l y a r i s e f r o m f i l l f l o w of t h e He 4 r i c h p h a s e , we have r e p e a t e d the e x p e r i m e n t s
w i t h e x t r a high p u r i t y He 3 , a n d w i t h He 3 c o n t a i n i n g 0.07% He 4 in w h i c h one w o u l d e x p e c t t h e p h a s e s e p a r a t i o n to o c c u r a t a b o u t 0.15OK 3). A l l t h e s e e x p e r i m e n t s g a v e s i m i l a r r e s u l t s , a s shown in fig. 2. L a n d a u ' s t h e o r y of a F e r m i l i q u i d 4) l e a d s to t h e c o n c e p t of a r e l a x a t i o n t i m e T c h a r a c t e r i s t i c of t h e c o l l i s i o n s b e t w e e n He 3 a t o m s (or m o r e a c c u r a t e l y b e t w e e n q u a s i - p a r t i c l e s ) w h i c h give r i s e to t h e v i s c o s i t y . A n a l y s i s of t h e v i s c o s i t y and o t h e r t r a n s p o r t p r o p e r t i e s s h o w s t h a t T i n c r e a s e s r a p i d l y with f a l l i n g t e m p e r a t u r e and a t 0 . 0 9 ° K h a s a v a l u e s u c h t h a t o~T ~ 1, w h e r e o~ i s the a n g u l a r f r e q u e n c y of t h e 1000 M c / s sound w a v e . H e n c e t h e a b r u p t c h a n g e in t h e v a l u e of Z/p o c c u r s when t h e m e a n f r e e p a t h b e t w e e n collisions becomes comparable with the wave length of t h e sound. T h e o n l y t r e a t m e n t of t h e p r o p a g a t i o n of sound in He 3 a t v e r y low t e m p e r a t u r e s i s t h a t of L a n d a u 5) b a s e d on h i s t h e o r y of a F e r m i l i q u i d . B e c a u s e of t h e long r e l a x a t i o n t i m e b e t w e e n c o l l i s i o n s , a sound w a v e c a n n o t p r o p a g a t e in the n o r m a l w a y , b u t L a n d a u p r e d i c t s t h a t t h e r e w i l l be a new c o l l e c t i v e m o d e of p r o p a g a t i o n . T h i s new m o d e (which he t e r m s z e r o sound) i s c h a r a c t e r i s e d b y a d i f f e r e n t v e l o c i t y a n d a d i f f e r e n t c o e f f i c i e n t of a b s o r p t i o n , but t h e s e q u a n t i t i e s a r e v e r y d i f f i c u l t to m e a s u r e on a c c o u n t of t h e high f r e q u e n c i e s and low t e m p e r a t u r e i n v o l v e d . H o w e v e r , i t a p p e a r s t h a t in the present experiment we are observing the trans i t i o n f r o m o r d i n a r y to z e r o sound. In the r e g i o n of z e r o sound t h e c l a s s i c a l r e l a t i o n b e t w e e n a c o u s t i c i m p e d a n c e and v e l o c i t y of sound ( Z = pc) i s no l o n g e r v a l i d . T h e r e f o r e w e c a n n o t
SHAPE
RESONANCES IN SIMPLIFIED
LETTERS
1 June 1963
i n t e r p r e t t h e v a l u e s of Z/p b e l o w the t r a n s i t i o n d i r e c t l y a s the v e l o c i t y of z e r o sound (Co). E x a c t a l g e b r a i c e x p r e s s i o n s g i v i n g the r e l a t i o n b e t w e e n Z / p and co h a v e b e e n g i v e n b y B e k a r a v i c h and K h a l a t n i k o v 6) in c o n n e c t i o n w i t h t h e i r c a l c u l a t i o n of t h e t h e r m a l b o u n d a r y r e s i s t a n c e b e t w e e n s o l i d s a n d l i q u i d He 3. A t p r e s e n t M r . G. A. B r o o k e r of t h i s l a b o r a t o r y i s e n g a g e d in t h e l e n g t h y p r o c e s s of o b taining numerical values for these expressions, using the latest values for.the various parameters c h a r a c t e r i s i n g t h e l i q u i d 7). So f a r , i t a p p e a r s t h a t f o r e x c i t a t i o n s in the liquid w h i c h a r e s p e c u l a r l y r e f l e c t e d a t the b o u n d a r y , Z/O i s of the o r d e r of c o. H o w e v e r , w h e t h e r Zip i s l e s s t h a n o r g r e a t e r t h a n c o a p p e a r s to d e p e n d r a t h e r c r i t i c a l l y on the v a l u e s of t h e a b o v e m e n t i o n e d p a r a m e t e r s . F u l l d e t a i l s of all this work will be published subsequently. W e a r e m u c h i n d e b t e d to t h e D . S . I . R . f o r a g r a n t f o r e q u i p m e n t , f o r t h e s u p p o r t of one of u s ( P W M ) , and f o r a r e s e a r c h s t u d e n t s h i p (BEK).
References 1) E.H.Jacobsen, Quantum electronics, ed. Townes (Columbia University P r e s s , New York, 1960) p. 468. 2) B. E. Keen, P.W. Matthews and J.Wilks, l>roc, of the VIIIth Int. Conf. on Low Temperature Physics, In p r e s s . 3) D.O. Edwards and J. G. Daunt, Phys. Rev. 124 (1961) 640. 4) L.D. Landau, J. Exptl. Theoret. Phys. (USSR) 30 (1956) 1058; translation: Soviet Phys. JETP 3 (1957) 920. 5) L.D. Landau, J. Exptl. Theoret. Phys. (USSR) 32 (1957) 59; translation: Soviet Phys. JETP 5 (1957) 101. 6) I. L. Bekaravich and I. M. Khalatnikov, J. Exptl. Theoret, P h y s . 39 (1960) 1699; translation: Soviet Phys. JETP 12 (1961) 1187. 7) D.Hone, Phys. Rev. 125 (1962) 1494.
SUPERCONDUCTORS THEORY *
-
II
C. J. T H O M P S O N a n d J. M. B L A T T
Applied Mathematics Department, University of New South Wales, Kensington, Australia Received 23 April 1963
In an e a r l i e r c o m m u n i c a t i o n 1) we r e p o r t e d t h e r e s u l t s of a c o m p u t e r c a l c u l a t i o n f o r a s u p e r c o n ducting uniform membrane, showing pronounced r e s o n a n c e s of t h e e n e r g y gap a s a f u n c t i o n of t h e width a of t h e m e m b r a n e . In o r d e r to g a i n a b e t t e r * This r e s e a r c h has been supported in part by U. S. A i r Force Research Grant No. 62-400 to the University of New South W a l e s .
i n s i g h t into the r e l e v a n t f a c t o r s , we h a v e p e r f o r m e d an a n a l y t i c c a l c u l a t i o n u n d e r c e r t a i n , r e a s o n a b l e , s i m p l i f y i n g a s s u m p t i o n s . In t h i s l e t t e r we p r e s e n t t h i s w o r k . If t h e s l a b e x t e n d s in t h e x - d i r e c t i o n f r o m x = 0 to x = a , and if w e i m p o s e p e r i o d i c b o u n d a r y c o n d i t i o n s in the y - and z - d i r e c t i o n s , with p e r i o d i c i t y d i s t a n c e L, t h e b a s i c o n e - e l e c t r o n w a v e f u n c t i o n
V o l u m e 5, n u m b e r 1
PHYSICS
LETTERS
1 J u n e 1963
has the f o r m (spin functions omitted):
a
Ck(r) = un(x ) exp (ik2Y + i k 3 z ) l L ,
C n = K -1~_, C n , A n , f
it)
where k 2 and k3 a r e integral multiples of 2 y / L and, for the sake of simplicity, un(0 ) = un(a ) = 0. Except for unimportant details, the boundary conditions used do not affect the final results. The functions ~k m a y be taken to be solutions of a Schr~Minger equation -(~2/2M) V2~ok + V ( x ~ k = CkCPk ,
(2)
nt
where (k F = F e r m i momentum, p = usual nondimensional coupling p a r a m e t e r = J P E / V )
ck = r/n + (ti2/2ND(k 2 + k~) ,
and A n = a r c sinh ( k ~ c / C n )
for
=0
tion of an electron p a i r . A n d e r s o n ' s 2) suggestion that the pairing should involve time r e v e r s e d functions (n, k2, k3, l) and (n, -k2, -k3, ~) is c l e a r l y app r o p r i a t e here f r o m s y m m e t r y . The choice of a model interaction and associated reduced m a t r i x elements is somewhat m o r e delicate since a simple delta-function potential
v ( r i - rj) = - gS(r i - rj) ,
(5)
leads to divergent integrals in superconductivity theory. We follow Gor'kov 3) by imposing an e n e r gy-dependent cut-off on the m a t r i x elements, obtained by s t r a i g h t f o r w a r d calculation from (1) and (5). Letting/~ be the chemical potential, the m a t r i x elements in question a r e
N = 2 ~k hk , where h k is related to Ck-/~ and Ck in the usual fashion, 3,4). By p e r f o r m i n g the sums over k 2 and k 3 analytically, we obtain
~ Hn ,
(12)
1"l
I Ck,-~l < tZ~c,
otherwise.
Cn (independent of k2, k3) for I ck-~] < otherwise .
(II)
In p r a c t i c e , we a r e given the number density N / V , not the chemical potential ~. The relation between the two quantities is given by
N/V = (M/2yti2a) Hn =-2(rbw~)
= ~
~c, (7)
This r e p r e s e n t s a highly "anisotropic gap". In the usual integral equation 3,4) of superconductivity theory, one can p e r f o r m the sums over k 2 and k 3 analytically. The r e s u l t is
for
~n-# < - ~ c
,
- [ c.2 + (~c)21 ~ - (~.-.) + [ c~ + (~ _.)2]½ for I~.-~1 < ~c,
(6)
Although the integral depends on n and n' only, the m a t r i x elements t h e m s e l v e s depend also on k2, k3, k~ and k~ because of the cut-off condition. In the limit of large a, f o r m u l a (6) yields - J / L 2 a = - J / V within the allowed region for k ¢ k' (the only inter e s t i n g case). This is the model interaction used by Bardeen et al. 4). With m a t r i x elements of the f o r m (6), the energy gap function Ck b e c o m e s
=0
#v < ~ + ~ c < #v+l •
where
and
=
rb~-/~ > ~ v .
Although the sum over n' in (8) extends over all eigenvalues of eq. (4), we see f r o m (10) that it is actually a finite sum. The maximum value of n' contributing to (8) is the integer v defined by the condition
[Un(X)Un,(X)] 2 dx if both I ck - u l < PiCOc
=0
,
(10)
G o
Ck
for
(4)
In the superconducting state, the one-electron functions (I) a s s o c i a t e in p a i r s to f o r m a wave func-
- ( J / L 2) f
r/n-/~ < - ~ c ,
for It/n-/.ll < ~ c
(3)
- (~2/2M)(d2un/dx 2) + V(x)un(x) = ~nUn(X).
=
(9)
K = 2~tE2/JM = k F / ~ p
= -a ' a r c sinh ( ~ c / C n ) - ½ a r c sinh [(nn -/~)1 Cn ]
which r e d u c e s , together with (1), to
(8)
[un(x)un,(x)] 2{ix ,
0
=0
for
~?n-/~ > ~u~c .
(13) So f a r , our f o r m u l a e have been quite general. In the computer calculation, the potential V(x) in the SchrSdinger equation (4) was taken to be the e l e c t r o static energy of one electron in the e l e c t r o s t a t i c potential produced by the c h a r g e density of all the other electrons, plus a uniformly s m e a r e d out background positive charge. (4) was then solved, the m a t r i x elements were found f r o m (6), and eqs. (8) through (13) were solved with these m a t r i x elements. The computer then determined the new charge distribution, and iterated until a s e l f - c o n sistent solution was achieved. It was found, not s u r p r i s i n g l y , that the ultimate s e l f - c o n s i s t e n t solutions had nearly vanishing
Volume 5, number 1
PHYSICS
LETTERS
I
c h a r g e d e n s i t y t h r o u g h o u t t h e i n t e r i o r of t h e s l a b . T h e b o u n d a r y r e g i o n , in w h i c h C o u l o m b e f f e c t s a r e i m p o r t a n t , w a s found to h a v e a w i d t h "of a p p r o x i m a t e l y 2 Angstr{~m u n i t s , f o r r e a s o n a b l e p a r a m e t e r s . It i s t h e r e f o r e r e a s o n a b l e to ignore the potential
-5~0o[ k
V(x) altogether in order to gain a better qualitative insight. T h i s i s t h e b a s i s of t h e p r e s e n t c a l c u l a t i o n . W i t h V(x) = O, eq. (41 h a s t h e i m m e d i a t e s o l u t i o n Un(X I = (2/a)½ sin (n~rx/a) ,
(14) n=I,2,3~...
r~
---(h2/2M)(n'~/a) 2
1 June. 1963
,
,
(15)
which leads to the overlap integrals a
f
[un(X)Un,(X)] 2 dx = ( l / a ) ( 1 + ½ 5nn,).
(16) THICKNESS OF SLAB ( A U ) - -
o
Fig. 1. Chemical potential tL versus thickness of film. The cusps occur at the lower ends of the resonance r e gions, a is continuous a c r o s s the resonance r e gions and approaches the F e r m i energy eF for large thicknesses. The p a r a m e t e r used for this figure was: N/V = 2 x 1022 e l e c t r o n s / c m 3.
S u b s t i t u t i o n of (16) into (8) y i e l d s [1 - (An/2Ka) ] Cn = (1/Ka) ~, Cn, A n, n'=l = i n d e p e n d e n t of n,
(17)
w h e r e t h e u p p e r l i m i t v i s now g i v e n b y s u b s t i t u t i o n of ( 1 5 ) i n t o (ii):
t
1
v = integral part of {~z/~)[(2M/~2)(~+ ~ c ) ] ~ } .
(18)
A look at (10) s h o w s that A n is a function of C n only, except w h e n ]r~n-~] < ~c~c. F o r m o s t slab widths a, this latter case does not occur for any value of n (the n a r r o w regions of a in which this case does occur correspond to the exceedingly sharp resonance rises in Cn). If we r e s t r i c t our-
w
selves to the "ordinary" regions (no I??n-~[ s m a l l e r t h a n ~O~c), eq. (17) i m p l i e s t h a t Cn i s i n d e p e n d e n t of n , with the v a l u e i0
Cn = ~U~c/sinh [ga/(v+½)] =0
f o r n ..< v , n>/~
6a 3
(201
T h e p r a c t i c a l e v a l u a t i o n of e q s . (18), (191 and (20) s t a r t s f r o m a c h o i c e of t h e i n t e g e r v. By i g n o r i n g t h e s m a l l q u a n t i t y ~coc in (18), we c a n e l i m i n a t e ~ f r o m (18) and (20) t o g e t a l o w e r bound on t h e s l a b w i d t h a f o r t h i s v a l u e of u. If t h i s l o w e r bound i s d e n o t e d b y a p , a~ i s g i v e n b y = -~-~- ~p 3 - ~ '
Z~
(19)
.
v(v+½)(v+11} •
2O
THLCKNESS OF ~LAB ~,A.U)~
Eq. (12) c a n b e s o l v e d f o r # e x p l i c i t l y w i t h the r e sult
U = (~api2/uM) {--N V+~
i~
(21)
W i t h t h e c h o s e n v a l u e of v ~ - f o r m u l a (191 i s a p p l i c a b l e f o r s l a b w i d t h s in t h e r a n g e av+c(a) < a < au+ 1-c(a), w h e r e c(al i s a m e a s u r e of t h e s i z e of t h e r e s o n a n c e r e g i o n s . F o r e a c h v a l u e of a in the
Fig. 2. Superconducting energy gap p a r a m e t e r versus thickness of film. At each resonance a new value of n s t a r t s to contribute. The new values of n are shown on the figure. The peak heights lie well. above the bulk value C~. The troughs a r e only slightly below C=. The widths of the resonances range from ~ 0.04 A at the v = 3 resonance, to 0.1 A at the v = 8 resonance. The parameters used for this figure were: N / V = 2 x 1022 electrons/cm 3, p = 0.3, ~w c = 100OK.
r e l e v a n t r a n g e , we find the c h e m i c a l p o t e n t i a l f r o m (20), and t h e e n e r g y gap p a r a m e t e r f r o m (191. T h e r e s u l t s a r e shown in f i g s . 1 and 2 f o r t h e s p e c i a l case, ~oJc = 100OK, N / V = 2 × 1022 c m - 3 , and 0 = 0.3. F o r these values, ((a) ~ 0.02 a.u. for a = 8 a.u., and ((a I ~ 0.05 a.u. for a = 30 a.u. 11. T h e similarity of fig. 2 to the r e s u R s of the c o m puter calculation is striking. W e note f r o m fig. 1 that there is a secular rise in ~ as the slab is m a d e narrower. This is not a
Volume 5, number 1
PHYSICS
s p e c i a l e f f e c t in s u p e r c o n d u c t i v i t y , b u t o c c u r s a l s o in t h e n o r m a l s t a t e ( F e r m i s e a ) . T h e e n e r g y n e e d e d to p u s h one m o r e p a r t i c l e into t h e v o l u m e V = aL2 i s l a r g e r f o r a c o m p a r a b l e to a F e r m i w a v e l e n g t h , t h a n f o r a = L = VL T h e p e c u l i a r b e h a v i o u r of ~ a s a f u n c t i o n of a, s h o w n in f i g . 1, i s , h o w e v e r , e s s e n t i a l to o b t a i n t h e d e c i d e d a s y m m e t r y of t h e r e s o n a n c e s in f i g . 2. (If w e a s s u m e , i n c o r r e c t l y , t h a t / ~ equals the conventional Fermi energy ~2kF2/2M, t h e r e s o n a n c e p e a k s and t r o u g h s a r e found to b e a l m o s t s y m m e t r i c a l a r o u n d the b u l k v a l u e Coo of t h e e n e r g y gap). The narrow resonance regions can also be disc u s s e d b y a m o r e d e t a i l e d s o l u t i o n of (17) in t h e r e l e v .ant r e g i o n s . No new q u a l i t a t i v e p o i n t s e m e r g e . F i n a l l y , w e e x a m i n e t h e a s y m p t o t i c b e h a v i o u r of the e n e r g y gap and t h e width of t h e r e s o n a n c e r e g i o n s f o r l a r g e s l a b w i d t h s a ( i . e . , l a r g e v). F o r l a r g e v, (21) y i e l d s av ~ v ( # V I 3 N ) { = v ( ~ / k F ) .
(22)
T h e r e s o n a n c e s , t h e r e f o r e , o c c u r a t s l a b w i d t h s a, ectnal to an i n t e g r a l n u m b e r of half w a v e l e n g t h s ~ / k F of a n e l e c t r o n r i g h t a t the F e r m i s u r f a c e . Eq. (20) y i e l d s / ~ ~ ~ 2 k F 2 / 2 M = cF. T h e b u l k v a l u e of t h e e n e r g y gap i s coo
-
~-~c sinh (l/p) "
(23)
It is interesting to observe that the ratio Cn/Coo is independent of the cutoff energy ~.0c whenever (10) is valid, i.e., for all values of the thickness a other than the narrow regions corresponding to the resonance rises in C n. The strength pararneter p which does enter the ratio Cn/Coo measures the strength of the interaction right at the F e r m i surface. If w e define "peak" and "trough" values of the energy gap at resonance n u m b e r v through (cf.(19)) kFa v = I~/.~c]sL~ { ~ } ,
THE
CRITICAL
CURRENT
(24a)
OF
LETTERS
1June 1963
and
C T ,v = ~ c/sinh { _ ~ } ,
(24b)
respectively, and m a k e use of (22), the asymptotic forms of Cp,v/C~o and CT,v/Coo for large v are given, in the weak coupling limit p << 1, by
cp ,,,I
1
e..-'p
(2s)
and
1}
CT,v/Coo~e x p { -
p(2~-l)
(26)
"
T h e s e go to u n i t y f o r l a r g e s l a b t h i c k n e s s , t h e e x c e s s b e i n g i n v e r s e l y p r o p o r t i o n a l to t h e t h i c k n e s s . To o b t a i n t h e a s y m p t o t i c f o r m of t h e r e s o n a n c e w i d t h f o r l a r g e v, w e f i r s t o b s e r v e t h a t t h e u p p e r and l o w e r v a l u e s of a , a U a n d a L , a t r e s o n a n c e n u m b e r v, are defined by ~ - # = - ~o~c and ~v - /~ = + ~/COcrespectively. If w e use these facts together with (15) and the asymptotic value CF for #, w e get aU- a L~Pv
,
(2'/)
where B
~1
.V. 2M
F o r t h e p a r a m e t e r s u s e d in fig. 2, B ~ 1.2 x 10 . 2 A. Unlike t h e r a t i o Cn/Coo, a U - a L i s c u t - o f f d e p e n d e n t (and i n d e p e n d e n t of p). In f a c t , (2'/) s h o w s t h a t for large slab thickness, a U - a L is directly prop o r t i o n a l to ~ c . References 1) J . M . Blatt and C.J.Thompson, Phys. Rev. Letters 10 (1963) 332. 2) P.W.Anderson, J. Phys. Chem. Solids 11 (1959) 26. 3) L. P. Gor'kov, J. Exptl. Theoret. Phys. (USSR) 34 (1958) 735; translation: Soviet Phys. JETP 7 (1958) 505. 4) J. Bardeen, L.N. Cooper and J. R. Schrieffer, Phys. Rev. 108 (1957) 1175.
A SUPERCONDUCTOR
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R. A. K A M P E R C. E. R. L., Leatherhead Received 23 April 1963 T h e p u r p o s e of t h i s n o t e i s to r e p o r t an e x p e r i m e n t a l d e m o n s t r a t i o n t h a t a s u p e r c o n d u c t o r with a negative interphase surface energy cannot carry a n e t c u r r e n t in the p r e s e n c e of an e x t e r n a l m a g n e t i c f i e l d s u f f i c i e n t l y s t r o n g to p e n e t r a t e t h e i n t e r i o r
and c r e a t e t h e " m i x e d s t a t e " 1) w i t h o u t t h e a i d of s t r u c t u r a l i m p e r f e c t i o n s to s t a b i l i s e t h e c u r r e n t a g a i n s t t h e L o r e n t z f o r c e 2). If t h i s a s s e r t i o n i s t r u e then t h e c r i t i c a l c u r r e n t I c of a c y l i n d r i c a l s p e c i m e n of r a d i u s r in a t r a n s -