On the transition temperature of superconducting alloys - Biblioteka UMCS

ANNALES

.U N I V E R S I T A T I S MARIAE C U R I E - S K Ł O D O W S K A LUBLIN —POLONIA

VOL. XL/XLI, 41 SECTIO AAA 1985/1986

Instytut Fizyki UMCS

К. I. WYSOKIŃSKI, M. PIŁAT

On the Transition Temperature of Superconducting Alloys

O temperaturze przejściastopów nadprzewodzących

О -температуресзерхпроводящего перевод;, i сплавах

Dedicated

to Professer Stanisław Szpikowski on

occasion

of

his

60th

birthday

1. INTRODUCTION

The

study

of

the

superconductivity

in

disordered

systems

and particularly in alloys

has

its long

history [1"}.

The early approaches

of Anderson

and

Gorkov [2] have shown that nonmagnetic impurities

have a

little effect

on

the

transition temperature.

The

problem of

the

concentration

dependence

of

the transition temperature Te(x) remained

opened. The

experimental

data зло«

[з} a

variety

of

behaviours,

which roughly can be collected in

three

groups.

In the

first

group the transition temperature

Tc of an

А

ХВ1-Х

alloys interpolates linearly as

a

function

of

con­

516 К. I. Wysokiński, M. Piłat

centration x between

the

Tc

's

of

pure components T

c

(A

) and æ

c

(B). For other groups Tc

(x) possesses

a

minimum or a

maximum for

some concentration

0<x<1,

where the

T

c

( x ) £

Tc(B).

The

most

interesting

case

is

that

corresponding to

the

maximum.

Although the

effect

on the

relative scale

can be pretty

large -

transition

temperature

of

an

alloy

can

be two or three times

as large

as that

for pure

materials

- its value

is usually

small

of

order

of

few K.

There exists in the literature number

of

papers [4-73 de­

voted

to

the

study of the

.T

c(x).

Except

of

[41

all

they use weak

coupling

BCS

type

of

model interaction.

Previously (s]

we

have

formulated Eliashberg

type

of the theory

for

superconducting transition

metal

alloys

formulated

in Wannier representation.

Coherent potential approximation (CPA) has been used

to treat disorder.

That

approach was

based on

the

following

tight-binding Hamiltonian at a

fixed

configuration

of

ions

h =

не

+ Hion

+

H

e

_

ion (1

) where

H

e

=

2Z

+ 7

2. nj-ff

+ '

fci;ja

ie-

a

jff

i<r iff

ijo-

Hion

=

2

4 <3

>

i

13 «p

H

e

-ion=

’

2

X

4*

<U

i "u

?>iffa3ff (4) 13er

*

In the

above

formulae

nie

=

atç

aiff

and ai6-(

aj.ç) creates (an­

nihilates) the d

electron in

the Wannier state .with spin

6

,

the

t^j

are

hopping

matrix elements.

U^,

are

random

“energy levels”,

intrasite Coulomb

matrix elements

and

ion masses,

respectively,

u^ denotes the

Ct +h component

of the

displacement of an

ion of the 1

th site and

is

the

On the Transition Temperature of Superconducting Alloys 517

Slater

coefficient

describing

an

exponential

exp(-q°r)

decrease of the

d-electron

nave function

[9]. The presence of

the

atomic

parameters

of d electrons

in hamiltonian

(1)

is the

important feature of

the theory

[s]

because the various correlations

exist between

superconducting

and atomic

parameters

[31.

2. THE THEORY

In

this

section

we shall briefly

review

the

previous [81

theory.

The

starting

point

was the hamiltonian (1-5) defined

on a Cubic lattices.

Besides the usual Migdal

-Eliashberg type

of ap­

proximations [10] the another one has been

used. It

is

so

called

contact

approximation

in which

electron scattering processes

caused by

either

electron-electron or electron-phonon

interactions were

taken into account only if

the two

electrons

are

initially

both at the same site,

say

i

and

finally both at another site

say j

[4].

In the

tight binding scheme this

means

the

neglect of

all off-diagonal

(in

site indices) matrix elements of the

Greens

functions and

self-energies. The resulting Eliashberg equa

­

tions has been configurationally

averaged

by means of the

CPA

[11]

and

then solved

for

T

_c

.

The resulting

formula

for

the

transition

temperature is that

of McMillan

[12]

g 1.04(1 +-^eff)

T

= ~?

exp

- --- --Sii--- (6)

^eff "-Heff

<

1+

0,62

\sff

>

with the parameters pertaining

to

the Vi-x

aU

°y d

2

Aeff

=

KSA* I

хФд + <4 + WW

+

У NgDjj

[ У

+ (Чд +

Яв^дЗ 1

/®

(7)

P

*

ff =

-»W

+ -Heff111 <8)

518 К, I. Wysokiński, M. Piłat

where

f denotes the

averaged

hopping

matrix element,

0 is the

Debye temperature

of the

alloy,

d is the distance

between neighbouring

atoms

in a lattice, y =

1 - x.

N denote,

respectively,

the

partially and

totally averaged

electron densities

of

states at the

Fermi

level Е~ and

О

-Heff= (9)

2

,

Im Di ( w

*

io)

D.

= --b- \

dw --- =--- ----

—

-, i=A,B

(10)

i 1Г J co •

3.

CONCENTRATION DEPENDENCE OF T

_c

Equations (6)-(10)

form

a basis

for the

study of the con­

centration dependence

of Tc>

The problem

has

been

previously

studied numerically f5, 63

on

the

basis

of

similar

theories.

Here

we want to

study

the

concentration dependence of the tran­

sition temperature

without

numerical calculations. To

this

end let

us simplify

the expressions

for effective

parameters

and ji

e

f£(x). The

parameters

of

the

pure materials we

are here interested in are collected in Table 1.

Table

1. Some

of

the parameters

describing

pure superconducting

elements

[31.

I

! Ele J mart L___

Atomic

structure % (Г),

work J - ; m

func-

! e

!

T I

u,

ti

on ’ nr x* /г/x 1 1 / }e7J

!

(К)?

!

Ionic mass

(m.u.)

^Lattice

j ___ ____J

i

» Ti

i

i

Y

3d24s2

3d

3

4s2 0.93

3.40

; 380!

0.39

;

0.38;0.27 4.25 ;

390; 5.43

j 0.60;0.26

47.90

50.94

«

hex ; bcc

i i 1

!Zr

; Nb j LIO

•4d25s2

4d

4

5s1 4d

5

5s

1

i 0.91 I

î I fl

3.15

'_Il250! 0.53 _II

! 0.41 * 0.25

4.00 ; 275;

9.25 J 0.82;0.26

4.65

! 380} 0.92 t

0.38;0.20

i i

91.22

!

hex

1 92.91

;

bcc

95.94 '

bcc

!

}

Ta ; 5d

56s

2

1

——L_________

^{0.87 4.05 ; 225J}

4.48

j 0.65;0.21

11

■ 1 18(195

^{bcc i*}

On the Transition Temperature of Superconductions Alloys S19

For we

are

interested in qualitative dependence T

c(x)

rather

than

quantitative

one

we

can safely assume q

A =

4g

= q

0. Simila- rily

we neglect the

Of

dependence

in (7)

and (10).

These

lead to the expressions

>

erf

(x)

=

t

2

(x)[xN

A(x)5A

(x)

+

(1

-X)N

B

(X)

DB(xJ

(11) Jteff(X) «

[x

UaN2(x) +

(1

-

X) UB MB(x)] /II

(X)

Exact numerical

calculations in

principle

take

into

account

the

mutual

influence

of

electrons

on phonons. This

has

been

neglected in

(11).

The study

of the concentration dependence of any quantity encounters

one

obvious

problem. Kamely

the changes in

the posi

­

tion

of the Fermi level

with

concentration.

This

feature

makes

the

determination

of

the function

h

‘A,B^(x)

very difficult.

The

simplest

possibility

is that none

of

the

parameters

in (12)

is x dependent. Thus

we

get

3-eff(

x) s X

-^A +

“ x)-^b

(12)

J eff

(x) =

xji

A +

(1 -

x) ji

3

*

where the Л

А

^^» J

1/,

(g )

are x-independent parameters

charac

­

terising

pure elements.

This

virtual

crystal type of

approximation describes

quite

vieil

t.ie

æ

c

(x)

an æaxKb

i_

z alloy. The comparison

is

shown

in Figure

1.

Although the masses

of

Ta

and

Nb differ considerably

(see

Table 1

)

the

neglect of

the

concentration

dependence

of seem to play a

minor

role. This

may be related

to

the fact that

is

an integral

quantity - its concentration

dependence has

been

washed

out

by integration

and

also weighting

factor

(1/cJ )

in

(10).

So

we

retain this approximation and

try Пд(х)с»

x II®, N

B(x) (1

-

xJN®

where

M

A

and HB are to

be

understood as

the densities of states

at the

Fermi level

of

pure elements. Such

an approximation nicely describes

Tc

(x)

for lio^b^..

alloy as

seen

from Figure

2.

520 К. I. Wysokiński, M. Piłat

On the Transition Temperature of .Superconducting Alloys 521

Similarly

T

c(x) by

taking

N. (x) = fx

N?»

A

A

YNb. and_{Л l Л}

L?a, т Jk I

Jk can be

described

Then

leff (X) = x3/2 \ + (1 - x)3/2 Л3

yueff(x) = + (1 ' x)2 Ъ

(13)

The

resulting comparison is shown

in Figure 3.

The

most desirable type

of

dependence is

given in

Figure

4.

Here T

c

(x) of

an

alloy TixZr

is for 0.35 <

x

< 0.7 two or three

times that

of

pure elements. This can

be

described

by

assu­

ming

the

dependence

A

eff =

X

^A

+ (1 '

x>^3

2 2

(1^)

y

*

eff s

x Ai

+ <_

1 “ x)

Л5

522 К. I. Wysokiński, M. Piłat

Suggesting that the product

й^(х^(х)

is

concentration

inde­

pendent

while

Na(x)/N(x) should be proportional to

x.

Fig. 4.

The obtained

curve

Tc

(x) interpolates quite good

between

the

limits x

=

0

and

x

=

1. All the

parameters entering (6)

are realistic as they

are taken

from

experiment

and describe

the pure

elements. It

should

be

pointed

out

the variety of existing T

c

formulae [13] . We have used

here

the most popular one

and discus

­ sed the

To

(x

)

in terms

of

the effective electron-fonon coupling

^•eff

and Сои1О1пг>

Pseudopotential The

independent experi

­

ments

measuring

the concentration dependence

of Д. would be very desirable

as

an

independent check

of

the correctness

of

the

above

guess. (See however

[141 for

the examples of the concentration

dependence

of

the

(local

and total) densities

of

states

at

the

Fermi

level calculated in

CPA

for a

different

purpose).

On the Transition Temperature of Superconducting Alloys 523

3.

CONCLUSION

We

have analysed the

concentration

dependence of

the super

­

conducting transition temperature

of

substitutionally disordered

transition metal alloys. Using the

expressions

for

T

c(x)

obtained

'

previously [8']

we

tried to

extract the concentration

dependence

mainly of the electronic

density

of

states. Although

we have

guessed

(but

see

[141

) the explicit

form

of

this dependence rather

than obtained it from

the

solution of CPA

equations,

the conclusion

that

the

main concentration

dependence stems

from

the

electronic

densities

of

states’^

seems to be

valid.

The agreement of

the above values

of

Te(x) with

experimental

data is

as good as

that

[5,

6"]

obtained by

means of full numerical

analysis.

ACKNOWLEDGEMENT

The

authorswould

like

to

thank

dr A.

L, Kuzemsky for past

and

recent discussions. This

work

is partially

supported

by

INTIBS under

the contract CPBR 15.6/45.

REFERENCES

I. Gennes,

de,

P.

G.,

Superconductivity of metals and alloys)

V. A.

Benjamin

Inc., New

fork,

Amsterdam 1966.

2. A n d e

r s

о

n

P.

U., J. Phys. Chern. Solids 1959, 11.,

26ł

Gorkov L. P.,

Zh.

Eksp. Teor.

Fiz.

1959, 37, 1407.

3.

Vonsovskij

S.

V.,

I z j

u

m 0

v

Yu. A., K

u

r- m a e

v

E. Z.,

Superconductivity of transition metals

their alloys and

compounds,

Nauka, Moscow 1977 (in Russian).

4.

Kerker G.,

Bennemann K.

H., Sol. State Comm.

(1974),

15, 29.

5. Wein

к

auf A.,

Zittartz

J., Solid St. Comm.

1974, 14,

3651

J. Low

Temp.

Phys., 1975, IS,

229.

*

K.I.Ï7. take

the

opportunity to

thank

prof. J.

Appel

for asking him

that question

some

time

ago.

524 K. L Wysokiński, M. Piłat

6. К

о

1 1 e

y

E.

,

К

о

1 1

e y

W

.,

Beckmann

A., phys. stat.

sol.

(b) 1981, 110, 775.

7. A

p p

e 1

J.,

Phys.

Rev.

B,

1976, B15

,

5205;

8.

T a n k

e i

K., Takano F.,

Progr.

Theor.

Phys. 1974,

51,

938;

Dubovsky

0. A.,

Fiz. Tverd.

Tela 1978, 20, 5119.

8.

Wysokiński К. I.,

K

u

z e m

s k

y

A.

L.

J. Ion

Temp.

Phys.

1985, 52,

81;

Wysokiński

K.I., Annales UMCS, Sectio AAA, 1985,

38,

47.

9. Barisić

S., Labbe J., Friedel

J.,

Phys.

Rev.

Lett.

1970, 25,

919.

10.

Eliashberg

GM., Zh. Eksp. Teor. Fiz. 1960,

58, 966.

11.

V e

1 i

с

к

у В., Kirkpatrick

S.,

Ehren

­ reich

H.,

Phys. Rev.

1968,

175, 747.

12. Mc M

i 1 1

a n B. L.

Phys.

Rev.

1968,

167,

551.

15. Su

гиа

M.,

phys.

stat. sol. (b) 1985,

116

,

465;

1984, 121

, 209;

Allen P.

B., Mitrovic B.,

Sol. State Phys. 1982,

37, 1.

14. Kurata

У.,

J. Phys.

F, 1980,

10 ,1

STRESZCZENIE

Dla

szeregu

stopów metali

przejściowych obliczono

zależność temperatury przejścia

T

w

stan nadprzewodnictwa

od

koncentracji

atomów. Okazało się,

że

poprawny

opis,

funkcji Tc

(x)

dla stopów Ta

x

Nb1-x

’ Uo

xNb1-x

’

Vb1-x 1 V

xTa

x

uzyskuje si<?

uwzględniając

jedynie

zależność

gęstości stanów N^

(x

) oraz

Np(x) obliczonych w

ramach przybliżenia

potencjału koherentnego.

Opis zależności

1 x wymaga uwzględnienia zmian

gęstości

stanów

składu stopu. W pracy

zaniedbano

zależność od T

c(x) stopu TixZr

fononów

.ze

zmianą

x

innych

parametrów stopu.

О температуре сверхпооводяцсго перехода в сплавах 525

РЕЗЮМЕ

В

работе

вычислена температура сверхпроводящего

перехода ряда

сплавов.

Хорошие

согласие

с экспериментом получено

для

сплавов

Тах

лъ

1_х, Мо

^

ъ

1-Х

>

V

b

l-x

и v

x

Ta

i-x учитивая

за­

висимость

от

x только

плотности

электронных

состояний. Для

сплава

Tix

Zr1

_x

надо

также учить зависимость

от

х плотно

­

сти

фононных

состояний

и константы связи.