Effect of fluctuations on the magnetization of dirty superconductors

Solid State Communlcattons,

Voi

11, pp

1699-1702, 1972

Pergamon P r e s s

Printed m Great Britain

E F F E C T OF F L U C T U A T I O N S ON THE ~IAGNETIZATION OF DIRTY SUPERCONDUCTORS '~ Klaus D U s a d e l ' Laboratory of Atomic and Sohd State P h y s i c s , Cornell U m ~ e r s t t y , Ithaca, New York 14850

(Received 12 July 1972 by, P G de Gennes)

The s t a n d a r d e x p r e s s i o n for the f l u c t u a t i o n c o n t n b u t l o n to the free energ? zs modified m a way c o n s i s t e n t with the e l e c t r o n - p h o n o n model of s u p e r c o n d u c t i v i t y Using th~s mod~ftcatmn the magnet~zatlon is c a l c u l a t e d m the d~rty hm~t at T~

R E C E N T L Y there has beer: c o n s i d e r a b l e interest m the f l u c t u a n o n - m d u c e d m a g n e t i z a t i o n of superconductors abo~e the t r a n s l t m n temperature ir~ In the c l e a n hmlt a n d for not too dirty materials the experimental r e s u l t s 1 agree qmte well with t h e o r e t i c a l predictions z-4 However, the s i t u a t i o n is l e s s s a t i s f a c t o r y m the extreme d~rty hm~t While the numerical r e s u l t s of references 3 and 4 apparently d l s a g r e e wltn experiments, in the dirty hmlt theory b~ Mak~ and T a k a y a m a s a large conm b u t m n to the free energy is neglected which, ff t a k e n into a c c o u n t , modifies their result for the m a g n e t i z a t i o n s ~ g m f m a n t l y . T h e s e observat m n s led us to a new m v e s t l g a t m n of the magnetiz a t i o n m the extreme d~rty hm~t which ~s presented m th~s note

~

/~ ~ ( r , r ' ) = Z ,c' G ( r , r ' , ~ ) O ( r , r ' , - - ~ - -~),

(2) G are G r e e n ' s functlons for free e l e c t r o n s m the magnetic field B and the bar m e q u a t i o n (2) d e n o t e s the impurity a~erage The u n d e r l y i n g model of the metal is an e l e c t r o n - p h o n o n s y s t e m with the e l e c t r o n - p h o n o n i n t e r a c t i o n approximated by a p o m t - h k e retarded i n t e r a c t i o n and the retardation taken into account by r e s t r i c t i n g the -~-sum m e q u a t i o n (2) to v a l u e s of i'-v I s m a l l e r than the Debye-frequency ~z) However, e q u a t i o n ( I ) , whmh has been used previously, is then not q m t e correct within the model c o n s i d e r e d If we sum the ladder d~agrams m the e l e c t r o n - p h o n o n model, we have to e x c l u d e the term h n e a r m the c o u p h n g c o n s t a n t y b e c a u s e thls term corresponds to a p r o c e s s where a phonon with wave-~ ector q = 0 is exchanged and this ~s e x a c t l y zero Therefore, we have to modify e q u a t i o n (1), the modlfmatlon b e i n g

To begin with let us write down the wellknown e x p r e s s i o n for the contr~butmn to the free energy due to f l u c t u a t l o n s which ts obtained by summing the ladder diagrams

6.0_: = T

2

In (I- o ~ )

are the e l g e n v a l u e s of the integral operator

(i)

,v,

6ft = Z

2

[ln(1-g~)-g~]

(3)

The new term m e q u a t i o n (3), m the following referred to as the h n e a r term, does not c h a n g e the leading term c l o s e to the transltzon of the magnet~z a t : o n or of any other quantity whmh can be derived from ~ However, as we will s e e later, It chang~es the m a g n e t l z a t m n for finite B at T = T,: and tt is not n e g h g l b l e e v e n for very s m a l l B

'~ Work supported by the D e u t s c h e F o r s c h u n g s g e m e m s c h a f t and by the U S Office of Naval R e s e a r c h under Contract No NOOO14-67-A0077-0010, T e c h m c a I Report No 26 ' Permanent address and a d d r e s s from September, 1972 I n s t i t u t e fur T h e o r e t l s c h e P h y s l k , Umvers~tat Gottmgen, German~

1699

1700

THE ~IAGNETIZATION OF DIRTY SUPERCONDUCTORS

The h n e a r term has been d~scussed aireadF by Lee and PaFne ~ T h m r claim, however that this term ff summed over ,n and ,\ ~s proportional to the square of the e l e c t r o n dens~tF ~s correct only for a s t a t i c and p o m t - h k e l n t e r a c t m n b e t w e e n e l e c t r o n s As soon as r e t a r d a t m n effects are introduced, this sum can be shown to be d w e r g e n t and one has to d i s c u s s the h n e a r term w n h l n the e l e c t r o n - p h o n o n model to get a c o n s i s t e n t p:cture Non-local effects which g w e rise to a strong s u p p r e s s i o n of the m a g n e t l z a t t o n for m c r e a s m g B m the c l e a n h a l t 2 a should be unimportant m the extreme d*rty hm~t and therefore we may approximate ~,\~ by Y V' 1 "% 2~ - q,~

(4)

= ~,~

- 2eB(2n - 1))

U

(S)

-~

.

~_o

2~-+

~(B),

(11)

- ,~ [> ~(.,,m)]

~

--

"

\ [Y~ (" '')] [~-

(, m

-

I

i)~-,qj

4

(12)

we rewrite e q u a t m n (4) as

-z,C : 1

z

.

2 \

and

F(q:t~)

(7)

F@)

4< aft : T~ -_-.~/" in 2r~L. - 4<

>~(Lm) = ~[~-~- (my) 2]-

m?l,

>2(% m) = \ '[v2- (' ml ~- I)~/2] -(l m ! -

For materials for w M c h the ratio ~OD/2T~To iS not too large we may replace the ~-sum m equatmn (6) by ~ts f i r s t term U s i n g t M s approximahon and e x p r e s s i n g e q u a t i o n (3) in terms of this F we get

,n

(10)

"" [ ~e~ Ii\ [v, (v,,,)] ~(B):~"__ d~(l_e.)

and X r e p r e s e n t s the quantum numbers n, q and k for the L a n d a u l e v e l s Here and m the followmg we a s s u m e T = T: W~th the help of

g

869.

T. e3/2 ~ l 2

- D(q:

F(q):4,~T:

1

ol c~B

the followmg expressions

where 4',{ is given by 4~

11, No 12

In d o m g thls we consmer on[) 8.Q-, the contrP butlon to ~ Q for a hxed ~, for whlch the ,k-sum in equatmn (8) is cor~ergent Deforming then the contour to the real t-aMs we get a contrlbutlon to $Q,. w h m h depends on the m a g n e t m held and ~h~ch is well behaved ff s u m m e d over m and a second term mdependent of B Thls second term is Identlcal to /~f! ~(B = 0) and it is dwergent if s u m m e d o~ee m Ho-~e~er, as far as the magnetlza hon is concerned, we are only mterested in the m-sum of (3-0-,(B) - cS.Q~(B = 0) and thls sum is convergent After pertormlng the a - s u m m a t m n m the way described abo~e the q
~a~:4=L

\ol

(13)

The parameter > can be expressed zn terms of B and B:2(O), the cntmal held at zero temperature -, : 2~

2~To ) 2~To ~, 4,~

A.

(8)

T h i s e x p r e s s i o n for 6fi xs a c t u a l l y a dicergent q u a n t i t y . H o w e v e r , the part of ( ~ which d e p e n d s on B is c o n v e r g e n t . T h i s can be s e e n most e a s i l y by c o n v e r t i n g the sum over n into a c o n t o u r - i n t e g r a l w~th the help of the 'Ferm~-functlon ' (e '7= ~- 1)-~, /3 : r:(2eB)-'

(9)

I)/

Bo 2 (0)

1 78 - B

(14)

As one might expect m a dirty-hart theorg, the quantny ~(B) is a um~ersal functmn of the reduced magnetic held b = B/B~a(O ) Note that the tMrd term m the curly brackets m equation (12) is due to the hnear term m equatmn (8) It is instructlve to derwe analF~ically some hmltlng forms of the magnehzatlon. In the h m n B = O, only the hrst term for m = 0 m equation (12) contributes and we get v(B: 0):~

~(3/2)(I-~-~_],

(15)

Vol

11, No 12

T H E ' ~ I A G N E T I Z A T I O N OF DIRTY SUPERCONDUCTORS

Th~s result has been o b t a m e d prev~ousl~ by Patton, Ambegaokar and ~ 11klns s and a l s o numerlcall~ by Prange 7 For s m a l l B, we may expand ~(B) into an asymptotic ser~es and the following r e s u l t s are obtained W~thout the h n e a r term and b3 t a k m g onl~ the m = 0 c o n t r i b u t i o n we get to l e a d m g order ,(S)

~ ~(s

= o) -

6 ~ (2",)

(16)

wh~le by taking all m-terms into account but st~I1 leaving out the h n e a r term w e get

x ( B ) - r(S = 0)

~

1

6 ~(2v)

(17)

Therefore, the sum over m becomes ~ery ~mportant r a t h e d,rty hm~t even for s m a l l B However, the result (17) clearly is not what one would expect p h y s m a l l y If B ~s i n c r e a s e d , f l u c t u a t m n effects should d e c r e a s e and so should v(B) E v a l u a t i n g then the full e x p r e s s m n (12), we indeed get a d e c r e a s m g r(B)

x ( S ) - t(O)

.Ta

1 (~(3,/2)- 1 5) (18) 6 ~(~,)

T h e s e a n a l y t i c a l r e s u l t s for s m a l l B demonstrate c l e a r l y the importance of the h n e a r term m equatmn (8) ~n the extreme d~rty hm:t On the other hand, ~f this cond~tmn ~s not met, the h n e a r term plays only a minor role according to the n u m e r m a l c a l c u l a t i o n s of Lee and P a y n e ~ At th~s point we should mention that the method of Kurkuarvl et al 3 for e x t r a c t i n g the magn e t i z a t i o n from a divergent e x p r e s s m n of the free energy ff a p p h e d to our e x p r e s s i o n for $fl gives asymptotlc r e s u l t s for x ( B ) which agree completely w~th the ones d~scussed above The present theory is v a h d only for B ~< Bca (0), s i n c e ff B e x c e e d s Bo 2 (0) s l g m f l c a n t l y n o n l o c a l corrections to equation (4) become ~mportant A numermal c a l c u l a t i o n now shows that x ( B ) agrees extremely well with e q u a t m n

1701

(18) not only for small B but for fields B up to B~a (0) Therefore, our theory predicts that o~ er the whole range 0 ~< B < B~a (0), U is given b.~

J

(19) Expressed m terms of the susceptlblhty ,~ = ~I/B, the second term m equatlon (19) represents a constant paramagnetlc contrlbutmn from the first terms

At present there are no p u b h s h e d data on the m a g n e t t z a t m n for extreme dirty m a t e r i a l s In the d i r t i e s t sample of reference (I) the ratio of the c o h e r e n c e length ~o, ~:o = v,'2v-Tc , to the mean free path l is only about 3 and this is much too s m a l l for the extreme d~rt~ hmlt theory to be a p p h c a b l e We expect that farms ~:0/! of at l e a s t 50 are needed for the extreme dirty hmlt c o n d l t l o r to be e s t a b h s h e d To c o n c l u d e we would hke to add some further remarks on the present theory At the t r a n s i t i o n to the s u p e r c o n d u c t i n g s t a t e the sum of ladder diagrams becomes s i n g u l a r and thls g i v e s rise to the b e h e f that p h y s l c a l propertms of the s y s t e m are dominated by t h e s e p r o c e s s e s One might expect then that the completely i n n o c e n t - l o o k i n g term g ~'~ should be ent~rely n e g h g l b l e . However, our c a l c u l a t i o n shows that this is not the c a s e It rather s e e m s that only the l e a d i n g term of a p h y s l c a l q u a n t i t y c l o s e to the t r a n s i t i o n is correctly descrlbed by the s i n g u l a r c o n t r l b u t m n to the ladder diagrams while d e t a i l s of th~s q u a n t i t y away from the t r a n s i t i o n have to be c a l c u l a t e d with more care T h i s lmmedmtely r i n s e s the q u e s t m n whether diagrams other than the ladder d~agrams have to be r e t a i n e d m thls region This c e r t a i n l y ~s an i n t e r e s t i n g q u e s t i o n but we are not able to answer ~t at the present stage q c k n o w l e d g e m e n t s - I am indebted to Professor V Ambegaokar for s t i m u l a t i n g d i s c u s s i o n s

1702

THE MAGNETIZATION OF DIRTY SUPERCONDUCTORS

~ol

REFERENCES 1

GOLLUB J p , B E A S L Y M R and TINKHA%'! ~,{ , P&~a Re~

o

LEE P A

3

KURKIJARVI J , A.~IBEGAOKAR V and E I L E N B E R G E R G , Ptz~s Re~, BS, 868 (1972)

4

LEE PA

5

MAKI K and TAKAYAbIA H , J

6

P A T T O N B R , A~IBEGAOKAR V and V~ILKINS J , Solzd Stare Commua

7

PRANGE R E , Ph~a

and PAYNE M G , P k ~ s

and P A Y N E , ~ I G , P k > s

Rev

Re,

Re;

Lo~

Legt

f_.err 25, 1646 (1970)

26, 1537 (1971)

BS, 923 (1972)

-fe'~p

Pk~s

5, 13 (1971) 7, 1287 (1969)

B1, 2349 (1970)

Der bekannte Ausdruck fur den Fluktuatlonsbeltrag zur fremn Energle wlrd modlflzlert in emer ~else, dle mlt dem Elektron-Phonon ~lodell der Supralettung ~m Emklang Ist \ht Hllfe dmses neuen Ausdrucks wlrd d~e Magnetlsmrung !"~ sog dlrt} hmlt berechnet

11, No 12