Transition-metal impurities in a superconductor

Solid State C o m m u n i c a t i o n s ,

Voi

11, pp. 1 7 3 5 - 1 7 3 9 , 1972.

Pergamon Press

P n n t e d in Great Br i t ai n

T R A N S I T I O N - M E T A L I M P U R I T I E S IN A S U P E R C O N D U C T O R * Hlroyukl Shiba Department of P h y s l c s , F a c u l t y of S c i e n c e , O s a k a U m v e r s l t y , ToFonaka, J a p a n

(Reeetved 28 September 1972 by T ,Nagam,ya)

S u p e r c o n d u c t i n g d i l u t e a l l o y s c o n t a i n i n g t r a n s l t l o n - m e t a l Impurities are s tu d i e d within the H a r t r e e - F o c k approximation L o c a h z e d e x c i t e d s t a t e s In the e n e r g y gap. the d e c r e a s e of the t r a n s i t i o n temperature and the c h a n g e of the jump of the s p e c i f i c heat due to Impurities are discussed

MUCH WORK has b e e n d e v o t e d to e f f e c t s of Impurities on s u p e r c o n d u c t o r s ~ T r a n s l t l o n - m e t a l impurities are usuali> c l a s s i f i e d a c c o r d i n g to their nature into two c a t e g o r i e s , nonmagnetic and magnetic c a s e s There are two s e p a r a t e a p p r o a c h e s for t h e s e two c a s e s T h e A b r l k o s o v - G o r k o v theor~ z b as ed on the s - d i n t e r a c t i o n (Kondo H a m l l t o m a n ) is often applied to paramagnetlc ~mpuntles, w h i l e for nonmagnetic lmpurlt~es s u c h as iron group e l e m e n t s in aluminum the nonmagne ti c r e s o n a n c e orbital model 3 a is regarded as the most s t a t a b l e But it :s highly d e s i r a b l e to have a unified theory, which c o v e r s n o n m a g n e t i c c a s e s as well as magnetic c a s e s in the same framework T h e purpose of t h i s c o m m u n i c a ti o n is to p r e s e n t a simple theory m thls d i r e c t i o n

where dcr(d~) is the c r e a t i o n ( a n n l h l l a t l o n ) operator of an e l e c t r o n in the r e s o n a n c e orbital In e q u a t l o n ( 1 ) is the reduced pairing a 4 due to Cooper pmrlng of c o n d u c t i o n e l e c t r o n s For >:o, where the operator 4 ; is a vector ( a i , a ~ a ~ a~, ), and 4~ its c o n j u g a t e

Our model is the same as the R a t t o - B l a n d l n ' s T he host metal is d e s c r i b e d by the BCS Hamlltonlan, and t r a n s l t l o n - m e t a l impurities are assumed for slmphcltTy to be r e p r e s e n t e d bF the Anderson models with a s i n g l e orbital Throughout this paper a g e n e r a h z e d H a r t r e e - F o c k approximation is ap p l i ed In such a way that the c o r r e l a tion term in the Anderson H a m l i t o n l a n is r e p i a c e d by

Let us begin with the study of a s i n g l e impurity in a superconductor It is e a s y to show that the c o n d u c t i o n - e l e c t r o n G r e e n ' s function Oee,(co) has the form

c~,(~o)

=

c~(~,)<~, - c~(~)t(=) @(~) (2)

with the BCS G r e e n ' s function G~(CO) = [co - ~:kFa - AocraPz]-'

- Ud~dt

- Udfd

[

(1)

(3)

Here (z, and p~ are the P a u h matrix in spln space and in partlcle-hole space respectlveIy s In the

* A part of thls work was done durmR the a u t h o r ' s s t a y at the I n s t i t u t e for Solid State P h y s m s , UnlversltF of Tok?o, Tokyo

approxlmatlon (I) the t-matrlx is given by

t(~,) = ~' [ I 1735

O~(-~)

[]-',

(4)

1736

TRANSITION-\IETAL

I ' q P U R I T I E S IN % S U P E R C O N D U C T O R

v, here

~,ol

(n) Uc,#qerzc /,,~:~g ( v i

11, No

12

\:)

In t h i s l i m i t we eas~i'~ o b t a i n t h e L E S

<5) w i t h t h e s - d m ~ x m g !,, t h e r e d e f i n e d d - l e v e l e n e r g y E= = E l 6"2 ('~z:t',
_c,

,

=

Z _\ 5-'

\ [ ( E ,E_L - F ~ - ) : - [~:(E:, - E : ~ ) ~1 Ee-

-

E~ - L : n . _ _

w h m h r e p r o d u c e s t h e r e s u h for a c ! a s s l c a i s p i n m t h e s - d m o d e l s s w h e n E e - 0 In f a c t , p u t t i n g E= = 0 we o b t a i n

-v\~(O)

:

'

-

koc~a,-'~

(6)


\ I% - -~' F r o m equations (4)-(6) we o b t a i n

t(~)

:

k

2[ £

-

L 1

C/3~

3

2

\k-~

--

\

(7 3 --'3

T h e t - m a t r i x in

(8)

w h i c h c o r r e s p o n d to i o c a h z e d ( L E S ) m t h e e n e r g y gap (8) for h m l t m g c a s e s

-

(7)

w i t h 1~ = v ' ~ , ' a \ " ~ ( 0 ) F r o m t h e d e f m l t l o n t h e red u c e d p a i r i n g A ! c a n be w r i t t e n m t h e form A z =-6AD with6 = 1~10 e q u a t i o n (7) h a s p o l e s a t

-

excited states

Let us examine equation

(0 Nonmagnetzc hrntt (~ = O) S i n c e we c a n s a f e l y a s s u m e t h a t \ o <-< [ ' , we h n d t h a t t h e l o c a h z e d e x c l t e d s t a t e s a r e loc a t e d m c l o s e v i c i n i t y to t h e e n e r g y gap ~ : ~ z [ 1 - 2-v~A~,'~.(Oy (1 - (:5)~] H e r e N~(a0) Is t h e L o r e n t z l a n d e n s i t y of s t a t e s of t h e d - l e v e l s T h e p o l e s a r e n o t h i n g but w h a t M a c h l d a a n d S h l b a t a 7 p o i n t e d out P u t t i n g AoW~:(O)= I0 -3 "~ i0 -~ and <.5 = i0 w e hnd 2-2~ao '\:(0)2 (i - (5)2"" I0 -3, ~hlch is ver> small

\a: _

__

a/

q

w h i c h is e s s e n t i a l l y e q u i v a l e n t t o e q u a t i o n (2 6) of r e f e r e n c e 6 9 ] u d g l n g from e q u a t i o n (9), o n e c a n e v p e c t a p p r e c i a b l e d e ; l a t l o n f:om t h e A b r > k o s o v - G o r k o v t h e o r ~ ~ h e n > -~ "" 1 C a r e f u i l b ' e x a m i n i n g e q u a t i o n (8), w e c o n c l u d e t h a t '.~hen U,'[" i s t o o s m a l l for t h e I m p u r l b to be m a g n e t m , we h a v e d o u b l } d e g e n e r a t e L E S m t h e v m m l t y of t h e e n e r g ? gap %]qen /_ ~s i n c r e a s e d b e y o n d a t h r e s h o l d v a l u e /_ - , t h e i ' n p u r l t ~ b e comes magnetm m the Hartree-Fock sense and t h e d o u b l y - d e g e n e r a t e s t a t e s p h t s : n t o t~vo, o n e of w h i c h m e r g e s i n t o t h e c o n t i n u u m a b o ~ e t h e g a p edge, while the other stabs m the energb gap acc o r d i n g to e q u a t i o n (9) Let us turn to the h m t e - c o n c e n t r a t m n problem We a s s u m e t h a t t h e i m p u r i t i e s are d L s t r l b u t e d r a n d o m l b m s p a c e a n d t h a t t h e r e is no m a g n e t i c ord e r i n g a m o n g t h e m H e r e ~ e toilo,a a c o n v e n t i o n a l treatment s 10The averaged conductmn-electron G r e e n ' s f u n c t i o n in N a m b u s p a c e is r e l a t e d t o t h e s e l f - e n e r g y part b y

Voi. 11, No. 12

T R A N S I T I O N - M E T - ~ L D,1PURITIES IN A S U P E R C O N D U C T O R

0.¢(-)

[ O i - ' - ~_ (.o)1-'

=

(11)

T = - T~:

hm :~

w~th G ~ = [ - -

~ . P ~ - Acra~a]-' (A IS the average

order parameter of the s u p e r c o n d u c t i n g a l l o ) ) ~<hm the a p p r o x i m a t m n of r e f e r e n c e 6 the seKe n e r g y part v (.~) is g i v e n by

: c
:(_)

¢(:~)

'

)'

(12)

where c ~s the concentratmn of the impurities, and < >',denotes the average o~er the dlrectlon of m a g n e t m moments T h e 'effectlve potentla1' ~(a0) is the same as before, but O~(~) should be determined self-consmtently by

[

= 1 E

6;~(~)

:

1

where 6 = -

hm

:-o F

E~-V

~-3~-(w-v)-'-

- F)'-

- F)

]

]

-V)~] ~--'`~(~

-F) ~

(16) (A,~/k) and .~-~ = (Z'z - 1)~kT.~o

Tc~ _

I

cT~: Eet - E : ~

9(I-

L~f~0)

\,(0) [_

,-7

[

8kT~

[~,<01-

\:~(0)]

(14)

- -~ [fi(E:~)"~:t (0) 2 - B(E~, ) \ :. (0) a] , (17)

-

CV

>

~,_\,,

1

where \ : - - ( ~ ) ,s the denszty of s t a t e s of d e l e c trons ~ l t h spin G

N~ :(o J) -

(157 .~ - r -

r)

The summation must be cut off at Deb~e frequenc~ ~o Performing the summatmn over - we fred that

='~(0)_~

<

-

2~-'(=0o

\~(0) E : , - E : :

and lntrcducmg a quantity u by u = 7., ~, we eas fly obtain aJ

(

-

-"

- F)'-] ~- - 4:-:I - ~

c~..2 { E ~ - ~ ' - ~ - ( - ~

(13)

i<~,h]-',

[L, - 4 . : ~ -

,-:

[E{ - v ~ - ( : .

It is not d i f f i cu l t to s e t up s e l f - c o n s i s t e n c y e q u a t i o n s from (11)-(13) Putting 6~&)

~,~(0)

-

hm T : -

O~(~-')

"T~:V~--V "

=

cT~

!_ : 6

_

1737

1

F

(18)

(~ - E j ~ ) ~ - F 2 Th e q u a n t i t i e s 0, /3 and g'~ff are d e h n e d by

2F(~'-')u-3 \~-

/2 2

Here the quantlt~es A, Ae, and
transztzon temperature

0

-

1 [E:~ = ( E z~) \ : * (0) E,:+ - E,:.;,

-

E e , ~(E=+)

c~(z) = log

2eC~D \ (V2 - za) "c,kT~o \ [ ( V - ab)z ~- z z]

_F(tan-' z

B(z)

=

1-

Th e i m t l a I d e c r e a s e of the trans ttmn temperature due to impurltles Is d e f i n e d by hm

\:+(0)],

z ~

__z ), tan-' F -0

log wkToo \ [(F

-

a-,D) 2 -

Z 2]

2z (tan-' z _ tan-, z - - - - 3 - ~ ) , F

C--~0

V

V

-9

(T= - T o J / c T : : ,

and it is very s e n s i t , v e to the magnetic property of the impurity T h e r e f o r e it is worthwhile to pay s p e c i a l a t t e n t m n to the initial d e c r e a s e The c a l c u l a t i o n of T: is trl~ lal 2 Expanding the right hand side of e q u a t m n (15) m terms of A a n d . s u b s n t u t m g it into the d e f m m o n of the order parameter A, we obtain

= gr

U I 1-~-E~t_E~;

(19)

1738

T R A N S I T I O N - M E T A L IMPURITIES IN A SUPERCONDUCTOR

with E d e r ' s c o n s t a n t C In the n o n m a g n e t i c limit (; = 0) e q u a t i o n (17) is reduced to the result by R a t t o - B l a n d m , 3 T a k a n a k a _ T a k a n o a and Kz,vlZucherman ~* for F -'" _~ and b) K a l s e r 12 for - Q < . [" On the other hand, m the mag,aet~c limit the dominant term g

E:~ - E ~

\ , ( 0 ) E:~

=

-~

E~, 8AT:-~

-

1 -

-

IV,:÷ (0)

1

-

(1-

for t<
--

"

hm

VoI

11, No

In the magnetic hmlt ,~,e

: 3-(i->~)

T: - T-:

,ff2<\(3~ ~2i)

~,e¢ ( 0 ) ]

-

12

with AC3 the jump of t h e s p e c l h c heat of the pure metal and X(n) = ' =~(21 ~ 1) -~ , ~,~hlle in the D n o n m a g n e t m hm~t 1 1

(20)

?~)

8fT.D ~,~(0)

C~

I

_

_

_

=(E_~j i

-

6

(9% --

represents the classlcal spin effect s w h e n

E', = 0 Notice the Abr~kosov-Gorkov theoo, is a h m l t m g c a s e ot equation (19) for Ez = 0 and { . : ' ; F 13 E q u a t m n (20) ~s a u n l f l e d e x p r e s s m n for the initial d e c r e a s e , which covers the nonm a g n e u c and magnetic c a s e s w,thm the H a r t r e e Fock approximation

is obtained Careful examination of our HartreeFock theory shows a continuous change of C*

~ith t, 'F from the Abrlkosov-Gorkov
( U " " V) to the e s s e n t m l l y B c g - h k e beha'~lor ({. <- V) in contrast to reference 14

(2) The lump ot the spec,fzc heat at the transition temperature

D e t a i l s of the present c a l c u l a t m n s and disc u s s i o n s of some related s u b j e c : s ,v Iti be pubh s h e d soon m a separate paper

The jump of the specffm heat at To, ~C, is also s e n s i t i v e to the nature of impurities ~4 Unfortunately we do not have enough s p a c e to full.~ d l s c u s s the change of AC due to Impurltms Therefore we show the e x p r e s s m n for AC only

4 c k n o w l e d g e m e n t s - The author ~ould hke to e x p r e s s his thanks to P r o f e s s o r K Yoslda, Professor A Yoshlmor< Dr A Sakurm ard Dr K Yamada for their ~aluable d ~ s c u s s m n s and hospltahb

REFERENCES

1

See for i n s t a n c e Superconducnvzt3, (Edited by PARKS R D ) Marcel Dekker, New York (1969), and S u p e r c o n d u c n v t t y , Proc Advanced Summer Stud~ lnst McGlll Unlv , Montreal (Edited b~, ~ A L L A C E P R ) Gordon 8~ Breach, New York (1969)

2

ABRIKOSOV A A and GORKOV L P , SovLet Ph>s J E T P

3

RATTO C F

4

T A K A N A K A K and TAKANO F ,

5

ANDERSON P W , Phys

Rev

124, 41 (1961)

6

SHIBA H , Pros theor Phys

40, 435 (1968)

7

MACHIDA K and SHIBATA F , Pros

8

SAKURAI A , Pros

9

For U >> F , v is equal to U/2 Therefore F / v is nothing but ( - J / 2 } J = - 8 , v /U r a t h e present c a s e

a n d B L A N D I N A , Ph>s

theor Phys

Rev

12, 1243 (1961)

156, 513 (1967)

Pros theor Phys

theor P h s s

36, 1080 (1966)

47, 1817 (1972)

44, 1472 (1970) S=D s i n c e S = 1/2 and

10

Z I T T A R T Z J , BRINGER A and MULLER-HARTMANN E , Sohd State Commun

11

KIWI M and ZUCKERMANN M ] , P h y s

12

KAISER A B , J

Phys

C 3, 410 (1970)

Rev

164, 548 (1967)

10, 513 (1972)

Vol

11, No

12

T R A N S I T I O N - \ I E T A L EMPURITIES IN -k SUPERCONDUCTOR

13

Since the t r a n s v e r s e component of a magnetic moment ts ignored in the H a r t r e e - F o c k a p p r o x i m a t i o n e q u a t i o n (20) g ~ e s S-' i n s t e a d of S ( S - 1) ~n the AG theor~

14

MULLER-HARTMANN E and Z I T T A R T Z J , preprmt

On dtudle les p r o p r l e t e s s u p r a c o n d u c t r l c e s d e s a l h a g e s d f l u e s a~ec d e s Impuretds des metaux de t r a n s m o n dans l ' a p p r o x l m a t i o n de Hartree-Fock Ondlscuteetats locahsesdans labande mterdlte, le changement de temperature s u p r a c o n d u c t r l c e et le c h a n g e m e n t de la d l s c o n t m u l t e de ia c h a l e u r s p e c l f l q u e

1739