Fluctuations in the resistive transition in aluminum films I: Purity dependence

V o l u m e 32A. n u m b e r i

FLUCTUATIONS

PHYSICS

IN THE

LETTERS

27 J u l y 1970

RESISTIVE TRANSITION PURITY DEPENDENCE

IN A L U M I N U M

FILMS

I:

K. KAJIMURA and N. MIKOSHIBA

Eleclrotcchnical Laboratory. Tanashi, Tokyo 188, Japan Received 26 May 1970

E f f e c t s of s a m p l e p u r i t y on the e x c e s s c o n d u c t i v i t y of Al f i l m s w e r e m e a s u r e d . The e x c e s s c o n d u c t i v ity was s t r o n g l y d e p e n d e n t on the e l e c t r o n i c m e a n f r e e pa t h in c o n t r a s t with the t h e o r y of A s l a m a z o v and Larkin.

Many i n v e s t i g a t i o n s * about the fluctuation p h e n o m e n a in s u p e r c o n d u c t o r s , both e x p e r i m e n t a l and t h e o r e t i c a l , have been c a r r i e d out in r e l a t i o n to the e l e c t r i c a l conductivity of twod i m e n s i o n a l s u p e r c o n d u c t i n g f i l m s (d << ~ (T), d: the f i l m t h i c k n e s s , ~(T): the G L - c o h e r e n c e length), above the t r a n s i t i o n t e m p e r a t u r e , Tco All r e s u l t s in the c l a s s i c a l r e g i o n w e r e d e c r i b e d by a s i m p l e f o r m u l a :

I

I

~oo

I

RN=~274~ohm/~

~

I

6z50

OJ~

i 2.0

[R N - R(T)]/R(T) = cr'/crs = eo[ Tc/(T- Tc) ] ,

I

ALUMINUM

I

;

3.0 TEMPERATURE ( ° K I

(1)

4..0

Fig. 1. Temperature dependence ofcrN/(r'. w h e r e R(T) is the film r e s i s t a n c e at t e m p e r a t u r e T, R N the n o r m a l state r e s i s t a n c e , a ' the e x c e s s conductivity due to the fluctuations, (~N the n o r m a l state conductivity in the a b s e n c e of f l u c t u a tions, ~ 0 the constant. A s l a m a z o v and L a r k i n [2] f i r s t p r e d i c t e d eq. (1) and the constant, c AL 0 = = e 2 / ( 1 6 t/daN). In this l e t t e r we p r e s e n t e x p e r i m e n t a l r e s u l t s of the effect of p u r i t y on the r e s i s t i v e t r a n s i t i o n . Experimental details are described elsewhere. The n o r m a l state r e s i s t a n c e , R N was m e a s u r e d i n a p e r p e n d i c u l a r m a g n e t i c field in which the r e s i s t a n c e did not change b e t w e e n 2.0°K and 4.2OK. Most m e a s u r e m e n t s w e r e c a r r i e d out in an annealed m u - m e t a l s h i e l d within which the m a g n e t i c field e s t i m a t e d was l e s s than 0.01 Oe. A t y p i cal t e m p e r a t u r e dependence of r e s i s t i v e t r a n s i t i o n for A1 f i l m is shown in fig. 1. In this figure , the quantity ~N/Cr' is plotted a g a i n s t the t e m p e r a t u r e in o r d e r to e s t i m a t e T c and ( 0 ' by the use of eqo (1). The l i n e a r p a r t in this f i g u r e shows an a g r e e m e n t with the t h e o r e t i c a l p r e diction of t e m p e r a t u r e dependence, i . e . , ~'~: T c / ( T - T c) r a t h e r above T c, but the m a g nitude of the conductivity is s t r o n g l y dependent on the e l e c t r o n i c m e a n f r e e path in d i s a g r e e m e n t with the p r e d i c t i o n . The a n o m a l y n e a r T c • See ref. [1] for works in this area. 216

w as o b s e r v e d in al l s a m p l e s . This anomaly is not due to the s a m p l e inhomogeneity, n o r due to the t r a n s i t i o n f r o m 2- to 3- d i m e n s i o n a l b e h a v i o r b e c a u s e the c o h e r e n c e length at the t e m p e r a t u r e is an o r d e r of magnitude l a r g e r than the film t h i c k n e s s . In the i m m e d i a t e v i c i n i t y of Tc the c r i t i c a l fluctuations s i m i l a r to that found by M a r ~ e l j a et al. [3] w e r e o b s e r v e d . T h e s e b e h a v i o r s will be d i s c u s s e d in d e t a i l in the next l e t t e r s [6]. 10

I

I

I

8 ALUMINUM o

g

o o

%°~

o o

+

2 AL THEORY

0 0.1

I' 1o 100 ' EFFECTIVE ELECTRONIC MEAN F R E E P A T H (~,)

1000

F i g . 2. E l e c t r o n i c m e a n A f r e e p a t h d e p e n d e n c e of E o b s / E ~ L . + r e f . [81 v

v

PHYSICS

Volume 32A. number 4

27 July 1970

LETTERS

Fig. 2 shows _ . t heel e c t r o n i c m e a n f r e e path dependence of e~bs/E~Lo The AL t h e o r y p r e d i c t e d that the quantity should be 1.0 i r r e s p e c t i v e of s a m p l e m e a n f r e e path. This e f f e c t c a n not be e x p l a i n e d by the n o n l i n e a r conduction t h e o r y [4, 5]. In fact, we d e t e r m i n e d E0 in the t e m p e r a t u r e r e g i o n w h e r e the n o n l i n e a r i t y

[I -cr'(E)/~'(0)] < 10-3, a'(E) and ~'(0) being the excess conductivity at the electric field, E, used for the resistance measurement, and the one at zero field limit respectively° After this work was completed, we noticed similar results obtained recently by Masker et al. [7]. Their results seem to agree with ours.

References [1] L.R. Testardi et al.. Phys. Rev. 181 (1969) 800. [2] L. G. Aslamazov and A. I. Larkin. Phys. Letters 26A (1968) 238. [3] S. Mar6elja, W.E. Masker and R. D. Parks. Phys. Rev. Letters 22 (1969) 124. [4] A.Schmid, Phys. Rev. 180 {1969) 527. [5] T.Tsuzuki. Phys. Letters 30A (1969) 285. [6] K. Kajimura and N. Mikoshiba, to be published. [7] W. E. Masker and R. D. Parks, Phys. Rev.. to be published. [8] M. Strongin et al., Phys. Rev. Letters 20 {1968) 922.

* * * * *

THE

KNIGHT

SHIFT

IN MAGNESIUM

K. CHHOTRAY and P. K. MISRA Department of P h y s i c s , Utkal University, Bhubaneswar-4, Orissa , India Received 24 June 1970

We have calculated the exchange enhanced spin susceptibility in magnesium and used this result to calculate the Knight shift. Our result indicates that other mechanisms which are usually neglected are important for magnesium.

T h e r e is l a r g e d i s a g r e e m e n t between the t h e o r e t i c a l [1] and e x p e r i m e n t a l [2] v a l u e s of the Knight shift in m a g n e s i u m . H o w e v e r , s i n c e t h e r e is l a r g e u n c e r t a i n t y in the t h e o r e t i c a l v a l ue, no meaningful explanation f o r this d i s a g r e e m e n t could be given. In this p a p e r , we r e p o r t the r e s u l t s of our c a l c u l a t i o n of the Knight shift in m a g n e s i u m with a m o r e r e l i a b l e method. The c a l c u l a t i o n of the Knight shift is s t r a i g h t f o r w a r d . The e x p r e s s i o n f o r the Knight shift is [3,4] KS = I n X l ~ o ( S d + S c p )

(1)

w h e r e ×~ is the spin s u s c e p t i b i l i t y p e r unit v o l ume ( i n ' c g s v o l u m e units) which m u s t include the effect of e l e c t r o n - e l e c t r o n i n t e r a c t i o n on the r e s p o n s e of e l e c t r o n s to a m a g n e t i c field, ~2o is the v o l u m e of the W i g n e r - S e i t z c e l l o v e r which the conduction e l e c t r o n wave function is n o r m a l i s e d , Sd is the a v e r a g e d d i r e c t spin d e n s i ty at the n u cl eu s due to the conduction e l e c t r o n s at

the F e r m i s u r f a c e and Scp is the spin density at the n u cl eu s f r o m the exchange p o l a r i s a t i o n of c o r e e l e c t r o n s in the p r e s e n c e of a m a g n e t i c field. Using this e x p r e s s i o n , J e n a et al. [1] have r e c e n t l y c a l c u l a t e d the Knifht shift in m a g n e s i u m to be 0.0554% while the e x p e r i m e n t a l value [2] is (0.1127 ± 0.0005) %. H o w e v e r , as the a u t h o r s have pointed out, t h e r e a r e i n a c c u r a c i e s in t h e i r c a l c u l a t i o n due to u n c e r t a i n t i e s in using a r e l i a ble v al u e f o r X* f o r which no e x p e r i m e n t a l v al u e P , is a v a i l a b l e . J e n a et al. [1] have c a l c u l a t e d Xp using S i l v e r s t e i n ' s e x p r e s s i o n [5] which is o b tained in an e f f e c t i v e m a s s a p p r o x i m a t i o n and the validity of which is not c l e a r f o r m a g n e s i u m s i n c e t h e r e a r e e s s e n t i a l l y two bands at the Fermi surface. We have c a l c u l a t e d the exchange enhanced spin s u s c e p t i b i l i t y in the following way. Th e t o tal m a g n e t i c s u s c e p t i b i l i t y , ×T, is given by XT

=

Xi + Xd + Xp

(2)

217