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Superconductivity of nonsubstitutional solid solutions: Pb(Ag) and Pb(Au)
SUPERCONDUCTIVITY
OF NONSUBSTITUTIONAL Pb(Ag) AND Pb(Au)*
SOLID
SOLUTIONS:
R. RAY,~ S. H. HAHN-~ and B. C. GIESSEN t Metastable thin films consisting of Pb (Ag) and Pb (Au) solid solutions with up to 12 at. ~o solute have been prepared by very rapid quenching from the melt (splat cooling). These solid solutions are not of the common, single-substitution type. The variation of the superconducting transition temperature T~ with composition x has been determined. A linear decrease of T~ with x was observed. By comparing the valence effects of T c in Pb(Ag) and Pb(Au) with those in other stable Pb solid solutions, an interstitial model was ruled out. The data support a di-substitutional model in which two solute atoms substitute i~or one solvent atom. SUPRACONDUCTIVITE DES SOLUTIONS SOLIDES Pb(Ag) ET Pb(Au) QUI NE SONT PAS DU TYPE SUBSTITUTIONNEL SIMPLE Des films minces mttastables eonstituSs par des solutions solides Pb(Ag) et Pb(Au), avec 12% at. de solut6 au maximum, ont @t6pr@par@spar trempe ultra rapide &partir du liquide (refroidisscment par projeetio~ et 6erasemcnt). Cos solutions solides ne sent pas du type usuel de substitution simple. La variation de la temp@rature de transition de supraeonduetivit6 T c avee la composition x a 6t6 d@termin@e. Une d6croissance lin6aire de Tc avec x a 6t6 observ6e. Si on compare los effets de valence de Tc dans Pb(Ag) et Pb(Au) & eeux des autxes solutions solides stables de Pb, une interpr6tation Kpartir de l'interstitiel est exelue. Los r6sultats eonfirment un mod~le de di-substitution dans lequel deux atomes de solut6 se substituent ~ un atome de solvant. SUPRALEITUNG IN DEN /qICHT-SUBSTITUTIONELLEN LEGIERUNGEN Pb(Ag) UND Pb (Au) Durch sehr sehnelles Abktihlcn aus der Schmelze (Klatsehkiihlung) wurden metastabile Pb(Ag)- und Pb(Au)-Schichten mit bis zu 12 At.~o Ag bzw. Au hergestellt..Diese Legierungen sind nicht vom normalcn Typ der Einfaeh-Substitution. Die Variation der Ubergangstemperatur T c mit der Zusammensetzung x wurde bestimmt. Es wurde eine lineare Abnahme von Tc mit zunehmendem x beobachtet. Dureh Vergleieh der Valenzeffekte von T~ in Pb(Ag) und Pb(Au) mit denjenigen in andercn stabilen Pb-Legierungen kann ein Zwischengitteratom-Modell ausgeschlossen werden. Die Daten spreehen fiir oin Substitutionsmodell, bei dem tin Matrixatom durch zwei Fremdstoffatome substituiert ist. 1. I N T R O D U C T I O N The m a x i m u m e q u i l i b r i u m solid s o l u b i l i t y of Ag a n d A u i n P b is ~ 0 . 2 a t . ~ . I t h a d 10ng been concluded from the high diffusion rates of A u i n P b t h a t the small gold a t o m s m u s t , a t least i n part, occupy i n t e r s t i t i a l sites r a t h e r t h a n replace lead atoms s u b s t i t u t i o n a l l y i n a 1:1 r a t i o ; (z-a) however, the low solid solubility has u n t i l n o w p r e v e n t e d m e a s u r e m e n t s of b u l k properties, such as density, of Pb(Ag) or P b ( A u ) . C o n c e n t r a t e d , m e t a s t a b l e metali n - m e t a l solid solutions w i t h solute/solvent size ratios r ( s o l u t e ) / R ( s o l v e n t ) ~ 0.7 have r e c e n t l y been prepared b y v e r y r a p i d q u e n c h i n g from t h e melt (splat cooling). (4) Y(Cu) solid solutions w i t h u p to 15 a t . ~ Cu were f o u n d to be i n t e r s t i t i a l , (5) while Gd(Fe) a n d Gd(Cu) solid solutions w i t h > 1 0 a t . ~ solute were d i - s u b s t i t u t i o n a l (spht interstitial). (6) These results followed from X - r a y d a t a a n d d e n s i t y m e a s u r e m e n t s ; t h e n o n - s u b s t i t u t i o n a l solid solutions have n e a r l y c o n s t a n t lattice parameters. Some concentrated, m e t a s t a b l e lead-base solid solutions w i t h silver a n d gold h a v e n o w b e e n p r e p a r e d ; t h e i r n e a r l y c o n s t a n t lattice p a r a m e t e r s i n d i c a t e t h a t t h e y are n o n - s u b s t i t u t i o n a l . The s u p e r c o n d u c t i n g
t r a n s i t i o n t e m p e r a t u r e s T c of m e t a s t a b l e Pb(Ag) a n d P b ( A u ) alloys w i t h u p to 12 a t . ~ solute are r e p o r t e d here. 2. EXPERIMENTAL PROCEDURE AND R E S U L T S
Alloy p r e p a r a t i o n b y splat q u e n c h i n g to 77°K a t cooling rates of ~ 10S°K/sec was as described i n Refs. 4 a n d 7. M e t a s t a b l e t h i n fihns of ~ 5 # t h i c k n e s s consisting of P b - b a s e solid solutions w i t h < 12 at. Ag a n d ~ 10 at. ~ A u were produced, c o r r e s p o n d i n g to a m a x i m u m increase of t h e solute c o n c e n t r a t i o n s over t h e e q u i l i b r i u m values b y a factor ~ 5 0 . I n separate X - r a y diffraction e x p e r i m e n t s it was established t h a t a t 7 7 ° K t h e P b - b a s e m e t a s t a b l e phases will n o t decompose i n t o the e q u i l i b r i u m phases. M a i n t a i n i n g this t e m p e r a t u r e , t h e t h i n films were t r a n s f e r r e d t o g e t h e r w i t h t h e copper s u b s t r a t e s t o a t t e - c r y o s t a t ; this p r e v e n t e d t r a n s f o r m a t i o n of the m e t a s t a b l e alloys. T c was d e t e r m i n e d b y m e a s u r i n g t h e resistance change of t h e t h i n films w i t h a f o u r - p o i n t probe i n c o r p o r a t e d i n t o the s u b s t r a t e , is) T h e observed t r a n s i t i o n w i d t h s d i d n o t exceed 70 m K . The r e s u l t i n g Tc d a t a are s u m m a r i z e d i n Fig. 1. The s c a t t e r of the t r a n s i t i o n t e m p e r a t u r e s b y 100 m K is well w i t h i n the limits of a c c u r a c y f o u n d i n other T~ studies on splat cooled foils. (9'1°) * Received May 1, 1972. Solid State Chemistry Laboratory, Department of T~ for several s u b s t i t u t i o n a l P b - b a s e solid solutions is Chemistry, Northeastern University, Boston, Massachusetts also g i v e n i n Fig. 1. 02115. ACTA METALLURGICA, VOL. 20, DECEMBER 1972 1335
1336
ACTA M E T A L L U R G I C A , VOL. 20, 1972 gr
]OC
,
;
,
;
.a
,
ISb . . . . . . . . . ~n C~.........
-IOC 0 ~
J
J
frequency of solute impurity modes vS should be considered instead of v D. Equation (3) has been used in discussing the valence effect. (12} Following Markowitz and Kadanoff, (13) one expects
J
0 Pb(Ag)
o
-20C
",
,-3OC ((3) i
20C
i
i',
.
".~ IOC
6 T c = ((~Tc)anis.-~- (6Tc)valenc e
""",(Au)
f
i
I
i
i
?
I
i
, . . . Bi
::::.--:
,', Pb(Au) (Pb)
............
-300
-400
,~ ,,~
-500
-600
~Au
-7o0 (b)
o
l
~'
=
~;
i
~ AT %
A i ;o SOLUTE i
~
s
i
,'2
FzG. 1. (~T~= Tc - - Tc(Pb) for metastable l~b(Ag) and Pb(Au) solid solutions. 6T~ for other Pb-base alloys (11~ also shown. (5 s, p) solutes in (a); (6 s, p) solutes in (b); Pb (Au) included in (a) for comparison. The changes 5T¢ occurring in a solid solution of concentration x follow straight lines of the form aT c
~T~ = ~ -~ ~
.x
(1)
with ~ --73 m K ; d T c / d x = --26 inK/at. ~ Ag for Pb(Ag) and T = --10 inK; d T d d x ~ --70 mK/at. ~ Au for Pb(Au). 3. D I S G U S S I O N
I n the following we will attempt to deduce from these data qualitatively the type of solid solution present in Pb(Ag) and Pb(Au), i.e. to decide whether they are interstitial or disuhstitutional. The BCS result for weak coupling superconductors k T c ~- hvj9 exp (--l/N(0) • V)
(2)
leads to (~T~ -T0
~vD -
-+ VD
1 - N(0)
(5[N(0)V] (3) V
(4)
i ~
J """
o
,
where the first term, due to smoothing of the gap anisotropy, predominates at small x, but saturates at ~ 1 - 2 at. ~ solute (is) and qualitatively explains the existence of the small, negative observed r values. The second term leads to the observed linear composition dependence of T c in (1). An evaluation of ((~Tc)valence based on the Eliashberg theory of electron-lattice interaction is due to Appel; (la) it takes the effect of impurity modes into account in detail but retains the linear dependence of (~Tc on the composition via the change of the pseudo-Coulomb potential U 1. I n the present case, no detailed information on the electron-phonon interaction exists; a qualitative explanation based on a comparison with the various slopes OTc/OX in Fig. 1 and an explanation in terms of (3) is therefore attempted. This approach m a y be justified by the observation that, for solute valences z ~ 4 i 1, the slopes O T c / a x are self-consistent and increase monotonically with the solute valence for the solutes in, Sn and Sb [Fig. l(a)]. A similar relation exists for the solutes T1, (Pb) and Bi [Fig. l(b)].(11) The data in Fig. 1 thus show that T~ varies as the valence electron concentration per primitive unit cell C~, with (~T~ > 0 for 5C~z > 0. For the present purpose it is sufficient as a zeroth approximation to consider (STd(~C~z to be independent of the solute species. CCt affects T c through the N(0)V term in (3); the detailed calculation (14) gives a change of U 1 leading to the same result. The small increase of T c at constant C~t for Pb(Sn) is due to the v~ term in (3). I t should be noted that a separation of N(0) and V in applying (2) to solid solutions is incorrect; for Pb as solvent, O N ( O ) [ O Q t < 0, (15~ consequently the N(0) term would give a decrease of T~ with increasing C~t. For non-substitutional solid solutions such as Pb(Ag), it is useful (6) to introduce a cluster size ratio Re, defined as the ratio of the number A - of vacated solvent atom sites to the number of added solute atoms B +, both taken per unit cell,
N(0)V
where vD ~ Debye frequency, N ( 0 ) ~ density of electron states at the Fermi surface and V - electron-phonon coupling parameter. I n comparing the effect of different solutes in the same solvent, the
R c = A - / B +.
(5)
R e varies from 0 for interstitial solid solutions (I) to 1 for substitutional solid solutions (S), with R e = ½ for disubstitutionals (S2, split interstitials).
R A Y et a l . :
SUPERCONDUCTIVITY
OF N O N S U B S T I T U T I O N A L
T h e d e p e n d e n c e o f C~ on t h e solute c o n c e n t r a t i o n x is t h e n given b y Ce~ = Z + (z - - R o Z ) "
x 1 +
(Re - - 1)x
(6)
where Z, z are t h e valences of s o l v e n t a n d solute, respectively. F o r P b ( A g ) a n d P b ( A u ) a t s m a l l solute contents, Z -----4, z = 1, x ~ 1 a n d (6) r e d u c e s t o : C~ ~-~ 4 + x
for i n t e r s t i t i a l s (I),
(7)
C~ ~ 4 - - x
for d i s u b s t i t u t i o n a l s ($2).
(8)
C~z in (8) is i d e n t i c a l to C~z for a s u b s t i t u t i o n a l P b solid solution w i t h a t r i v a l e n t solute, where R~ = 1, z ~ 3. To t h e e x t e n t t h a t C~ d e t e r m i n e s t h e second t e r m in (3) a n d hence ((~Tc)valence, t h e slope o f (~T~ for d i s u b s t i t u t i o n a l P b ( A g ) s h o u l d be n e g a t i v e , as it is for P b ( I n ) , a n d Pb(Cd) [Fig. l ( a ) ] a n d c o m p a r a b l e in m a g n i t u d e to t h a t of P b ( I n ) , as observed. F o r P b ( A u ) [Fig. l(b)], t h e slope is also n e g a t i v e , b u t its m a g n i t u d e far exceeds t h a t o f Pb(T1) (see below). F o r b o t h P b ( A g ) a n d P b ( A u ) , T~ does n o t increase as w o u l d be e x p e c t e d f r o m (7) for i n t e r s t i t i a l solid solutions. W e conclude t h e r e f o r e t h a t R~ ~ ½ for t h e s e alloys. T h e c o n t r i b u t i o n o f t h e i m p u r i t y frequencies t h r o u g h t h e first t e r m of (3) is discussed n e x t . I n P b alloys w i t h solute valences b e t w e e n 3 a n d 5, t h e slope of 5To is a l w a y s larger for t h e l i g h t e r o f t w o i s o v a l e n t solutes. (11) P b ( A g ) a n d P b ( A u ) follow t h i s rule [Fig. 1(a)]; t h e i n t e r a t o m i c force c o n s t a n t s K~ can be a s s u m e d t o be n e a r l y e q u a l for b o t h solutes a n d t h e frequencies of t h e i m p u r i t y m o d e s are t h e n a p p r o x i m a t e d b y t h e m a s s r a t i o s ( M i m p / M p b ) -1/2. H o w e v e r , t h e site g e o m e t r y suggests t h a t for n o n s u b s t i t u t i o n a l solutes such as A g or Au, K i increases over t h e v a l u e s in s u b s t i t u t i o n a l solid solutions. This is confirmed b y M o s s b a u e r r e s o n a n c e m e a s u r e m e n t s of t h e recoil-free f r a c t i o n f : for In(Co), where Co is a s s u m e d to be i n t e r s t i t i a l , (16) f is f o u n d t o be t h r e e to four t i m e s larger t h a n t h e values e s t i m a t e d for s u b s t i t u t i o n a l Co ; for Au(Co), t h e r e is a n increase o f f on passing f r o m a s u b s t i t u t i o n a l t o a dis u b s t i t u t i o n a l solid s o l u t i o n which c o r r e s p o n d s t o a decrease of t h e m e a n - s q u a r e d solute a t o m d i s p l a c e m e n t
SOLID SOLUTIONS
1337
b y 35 p e r cent. (17) T h e increase o f K i p r e s e n t in b o t h s o l u t i o n t y p e s will r e s u l t in localized h i g h - f r e q u e n c y m o d e s which raise T c A c o r r e s p o n d i n g , s u b s t a n t i a l increase o f Tc for i n t e r s t i t i a l solutes is d e r i v e d in A p p e l ' s analysis. (14) F r o m t h e s e c o n s i d e r a t i o n s we conclude t h e n t h a t t h e o b s e r v e d decrease o f T~ in P b ( A g ) a n d P b ( A u ) is n o t d u e to t h e f r e q u e n c y t e r m o f (3). I t m u s t t h e r e f o r e be d u e to t h e v a l e n c e effect t e r m t r e a t e d a b o v e which led to R~ ~ ½. This result agrees w i t h r e c e n t conclusions f r o m i n d i r e c t evidence (ls~ for t h e f o r m a t i o n o f A u d i - s u b s t i t u t i o n a l s (considered as d i - i n t e r s t i t i a l - v a c a n c y clusters, I 2 V ) in e q u i l i b r a t e d P b ( A u ) [from r e s i s t a n c e v a r i a t i o n s a n d p r e c i p i t a t i o n r a t e s of A u P b a in P b ( A u ) ] . I t w o u l d be o f i n t e r e s t t o s t u d y T c in o t h e r nons u b s t i t u t i o n a l m e t a l - m e t a l solid solutions, especially t h o s e w i t h R~--~ 0, w h e r e T c m a y increase s t r o n g l y on alloying. ACKNOWLEDGEMENT
S u p p o r t of t h i s w o r k b y t h e Office o f N a v a l R e s e a r c h u n d e r C o n t r a c t N14-68-A-207-3 is g r a t e f u l l y acknowledged. REFERENCES
1. W. SEITH and A. KEIL, Z. phys. 22, 350 (1933). 2. B. F. DYSON, T. R. ANTHONY and D. T~RNRULL, J. appl. Phys. 37, 2370 (1966). 3. T. R. ANTHONY, in Vacancies and Interstitials in Metals, edited by A. SEEGER. Wiley (1970). 4. B. C. GIESSEN and R. H. WILLENS, in Phase Diagrams, edited by A. M. ALPER, Vol. III, p. 103. Academic Press (1970). 5. B. C. GIESSE~, P~. RAY and S. i . HAE~, Phys. Rev. Lett. 26, 509 (1971). 6. R. RAY, M. SO~INI and B. C. GIESSEN, Solid State Commun. 10, 163 (1972). 7. C. BORROMEE--GAuTIER,B. C. GIESSEN and N. J. GRANT, J. chem. Phys. 48, 1905 (1968). 8. R. RAY and B. C. GIESSE:N,to be published. 9. C. C. TSUEI and L. R. NEWKIRK, Phys. Rev. 183, 619 (1969). 10. L. R. NEWKIR~ and C. C. TSUEI, Phys. Rev. B3,755 (1971). 11. E. NEMBACH,J. Phys. Chem. Solids 29, 1205 {1968). 12. D. M. GINSBERG, Phys. Rev. 138, A1409 (1965). 13. D. MARXOWlTZand L. P. KADA~OFF,Phys. Rev. 131, 563 (1963). 14. J. APPEL, Phys. Rev. 153, 421 (1967). 15. J. R. ANDERSONand A. V. GOLD, Phys. Rev. 139, A1459 (1965). 16. P. A. FLIN~, U. GO~SE~, R. W. GRANT and R. M. I-IOUSELY, Phys. Rev. 157, 538 (1967). 17. C. F. STEEX, D. G. HOWARD and R. H. NUSSBAU~, Solid State Commun 9, 865 (1971). 18. A. ROSSOI~IMOand D. TUR~BULL, Acta Met. in press.
OF NONSUBSTITUTIONAL Pb(Ag) AND Pb(Au)*
SOLID
SOLUTIONS:
R. RAY,~ S. H. HAHN-~ and B. C. GIESSEN t Metastable thin films consisting of Pb (Ag) and Pb (Au) solid solutions with up to 12 at. ~o solute have been prepared by very rapid quenching from the melt (splat cooling). These solid solutions are not of the common, single-substitution type. The variation of the superconducting transition temperature T~ with composition x has been determined. A linear decrease of T~ with x was observed. By comparing the valence effects of T c in Pb(Ag) and Pb(Au) with those in other stable Pb solid solutions, an interstitial model was ruled out. The data support a di-substitutional model in which two solute atoms substitute i~or one solvent atom. SUPRACONDUCTIVITE DES SOLUTIONS SOLIDES Pb(Ag) ET Pb(Au) QUI NE SONT PAS DU TYPE SUBSTITUTIONNEL SIMPLE Des films minces mttastables eonstituSs par des solutions solides Pb(Ag) et Pb(Au), avec 12% at. de solut6 au maximum, ont @t6pr@par@spar trempe ultra rapide &partir du liquide (refroidisscment par projeetio~ et 6erasemcnt). Cos solutions solides ne sent pas du type usuel de substitution simple. La variation de la temp@rature de transition de supraeonduetivit6 T c avee la composition x a 6t6 d@termin@e. Une d6croissance lin6aire de Tc avec x a 6t6 observ6e. Si on compare los effets de valence de Tc dans Pb(Ag) et Pb(Au) & eeux des autxes solutions solides stables de Pb, une interpr6tation Kpartir de l'interstitiel est exelue. Los r6sultats eonfirment un mod~le de di-substitution dans lequel deux atomes de solut6 se substituent ~ un atome de solvant. SUPRALEITUNG IN DEN /qICHT-SUBSTITUTIONELLEN LEGIERUNGEN Pb(Ag) UND Pb (Au) Durch sehr sehnelles Abktihlcn aus der Schmelze (Klatsehkiihlung) wurden metastabile Pb(Ag)- und Pb(Au)-Schichten mit bis zu 12 At.~o Ag bzw. Au hergestellt..Diese Legierungen sind nicht vom normalcn Typ der Einfaeh-Substitution. Die Variation der Ubergangstemperatur T c mit der Zusammensetzung x wurde bestimmt. Es wurde eine lineare Abnahme von Tc mit zunehmendem x beobachtet. Dureh Vergleieh der Valenzeffekte von T~ in Pb(Ag) und Pb(Au) mit denjenigen in andercn stabilen Pb-Legierungen kann ein Zwischengitteratom-Modell ausgeschlossen werden. Die Daten spreehen fiir oin Substitutionsmodell, bei dem tin Matrixatom durch zwei Fremdstoffatome substituiert ist. 1. I N T R O D U C T I O N The m a x i m u m e q u i l i b r i u m solid s o l u b i l i t y of Ag a n d A u i n P b is ~ 0 . 2 a t . ~ . I t h a d 10ng been concluded from the high diffusion rates of A u i n P b t h a t the small gold a t o m s m u s t , a t least i n part, occupy i n t e r s t i t i a l sites r a t h e r t h a n replace lead atoms s u b s t i t u t i o n a l l y i n a 1:1 r a t i o ; (z-a) however, the low solid solubility has u n t i l n o w p r e v e n t e d m e a s u r e m e n t s of b u l k properties, such as density, of Pb(Ag) or P b ( A u ) . C o n c e n t r a t e d , m e t a s t a b l e metali n - m e t a l solid solutions w i t h solute/solvent size ratios r ( s o l u t e ) / R ( s o l v e n t ) ~ 0.7 have r e c e n t l y been prepared b y v e r y r a p i d q u e n c h i n g from t h e melt (splat cooling). (4) Y(Cu) solid solutions w i t h u p to 15 a t . ~ Cu were f o u n d to be i n t e r s t i t i a l , (5) while Gd(Fe) a n d Gd(Cu) solid solutions w i t h > 1 0 a t . ~ solute were d i - s u b s t i t u t i o n a l (spht interstitial). (6) These results followed from X - r a y d a t a a n d d e n s i t y m e a s u r e m e n t s ; t h e n o n - s u b s t i t u t i o n a l solid solutions have n e a r l y c o n s t a n t lattice parameters. Some concentrated, m e t a s t a b l e lead-base solid solutions w i t h silver a n d gold h a v e n o w b e e n p r e p a r e d ; t h e i r n e a r l y c o n s t a n t lattice p a r a m e t e r s i n d i c a t e t h a t t h e y are n o n - s u b s t i t u t i o n a l . The s u p e r c o n d u c t i n g
t r a n s i t i o n t e m p e r a t u r e s T c of m e t a s t a b l e Pb(Ag) a n d P b ( A u ) alloys w i t h u p to 12 a t . ~ solute are r e p o r t e d here. 2. EXPERIMENTAL PROCEDURE AND R E S U L T S
Alloy p r e p a r a t i o n b y splat q u e n c h i n g to 77°K a t cooling rates of ~ 10S°K/sec was as described i n Refs. 4 a n d 7. M e t a s t a b l e t h i n fihns of ~ 5 # t h i c k n e s s consisting of P b - b a s e solid solutions w i t h < 12 at. Ag a n d ~ 10 at. ~ A u were produced, c o r r e s p o n d i n g to a m a x i m u m increase of t h e solute c o n c e n t r a t i o n s over t h e e q u i l i b r i u m values b y a factor ~ 5 0 . I n separate X - r a y diffraction e x p e r i m e n t s it was established t h a t a t 7 7 ° K t h e P b - b a s e m e t a s t a b l e phases will n o t decompose i n t o the e q u i l i b r i u m phases. M a i n t a i n i n g this t e m p e r a t u r e , t h e t h i n films were t r a n s f e r r e d t o g e t h e r w i t h t h e copper s u b s t r a t e s t o a t t e - c r y o s t a t ; this p r e v e n t e d t r a n s f o r m a t i o n of the m e t a s t a b l e alloys. T c was d e t e r m i n e d b y m e a s u r i n g t h e resistance change of t h e t h i n films w i t h a f o u r - p o i n t probe i n c o r p o r a t e d i n t o the s u b s t r a t e , is) T h e observed t r a n s i t i o n w i d t h s d i d n o t exceed 70 m K . The r e s u l t i n g Tc d a t a are s u m m a r i z e d i n Fig. 1. The s c a t t e r of the t r a n s i t i o n t e m p e r a t u r e s b y 100 m K is well w i t h i n the limits of a c c u r a c y f o u n d i n other T~ studies on splat cooled foils. (9'1°) * Received May 1, 1972. Solid State Chemistry Laboratory, Department of T~ for several s u b s t i t u t i o n a l P b - b a s e solid solutions is Chemistry, Northeastern University, Boston, Massachusetts also g i v e n i n Fig. 1. 02115. ACTA METALLURGICA, VOL. 20, DECEMBER 1972 1335
1336
ACTA M E T A L L U R G I C A , VOL. 20, 1972 gr
]OC
,
;
,
;
.a
,
ISb . . . . . . . . . ~n C~.........
-IOC 0 ~
J
J
frequency of solute impurity modes vS should be considered instead of v D. Equation (3) has been used in discussing the valence effect. (12} Following Markowitz and Kadanoff, (13) one expects
J
0 Pb(Ag)
o
-20C
",
,-3OC ((3) i
20C
i
i',
.
".~ IOC
6 T c = ((~Tc)anis.-~- (6Tc)valenc e
""",(Au)
f
i
I
i
i
?
I
i
, . . . Bi
::::.--:
,', Pb(Au) (Pb)
............
-300
-400
,~ ,,~
-500
-600
~Au
-7o0 (b)
o
l
~'
=
~;
i
~ AT %
A i ;o SOLUTE i
~
s
i
,'2
FzG. 1. (~T~= Tc - - Tc(Pb) for metastable l~b(Ag) and Pb(Au) solid solutions. 6T~ for other Pb-base alloys (11~ also shown. (5 s, p) solutes in (a); (6 s, p) solutes in (b); Pb (Au) included in (a) for comparison. The changes 5T¢ occurring in a solid solution of concentration x follow straight lines of the form aT c
~T~ = ~ -~ ~
.x
(1)
with ~ --73 m K ; d T c / d x = --26 inK/at. ~ Ag for Pb(Ag) and T = --10 inK; d T d d x ~ --70 mK/at. ~ Au for Pb(Au). 3. D I S G U S S I O N
I n the following we will attempt to deduce from these data qualitatively the type of solid solution present in Pb(Ag) and Pb(Au), i.e. to decide whether they are interstitial or disuhstitutional. The BCS result for weak coupling superconductors k T c ~- hvj9 exp (--l/N(0) • V)
(2)
leads to (~T~ -T0
~vD -
-+ VD
1 - N(0)
(5[N(0)V] (3) V
(4)
i ~
J """
o
,
where the first term, due to smoothing of the gap anisotropy, predominates at small x, but saturates at ~ 1 - 2 at. ~ solute (is) and qualitatively explains the existence of the small, negative observed r values. The second term leads to the observed linear composition dependence of T c in (1). An evaluation of ((~Tc)valence based on the Eliashberg theory of electron-lattice interaction is due to Appel; (la) it takes the effect of impurity modes into account in detail but retains the linear dependence of (~Tc on the composition via the change of the pseudo-Coulomb potential U 1. I n the present case, no detailed information on the electron-phonon interaction exists; a qualitative explanation based on a comparison with the various slopes OTc/OX in Fig. 1 and an explanation in terms of (3) is therefore attempted. This approach m a y be justified by the observation that, for solute valences z ~ 4 i 1, the slopes O T c / a x are self-consistent and increase monotonically with the solute valence for the solutes in, Sn and Sb [Fig. l(a)]. A similar relation exists for the solutes T1, (Pb) and Bi [Fig. l(b)].(11) The data in Fig. 1 thus show that T~ varies as the valence electron concentration per primitive unit cell C~, with (~T~ > 0 for 5C~z > 0. For the present purpose it is sufficient as a zeroth approximation to consider (STd(~C~z to be independent of the solute species. CCt affects T c through the N(0)V term in (3); the detailed calculation (14) gives a change of U 1 leading to the same result. The small increase of T c at constant C~t for Pb(Sn) is due to the v~ term in (3). I t should be noted that a separation of N(0) and V in applying (2) to solid solutions is incorrect; for Pb as solvent, O N ( O ) [ O Q t < 0, (15~ consequently the N(0) term would give a decrease of T~ with increasing C~t. For non-substitutional solid solutions such as Pb(Ag), it is useful (6) to introduce a cluster size ratio Re, defined as the ratio of the number A - of vacated solvent atom sites to the number of added solute atoms B +, both taken per unit cell,
N(0)V
where vD ~ Debye frequency, N ( 0 ) ~ density of electron states at the Fermi surface and V - electron-phonon coupling parameter. I n comparing the effect of different solutes in the same solvent, the
R c = A - / B +.
(5)
R e varies from 0 for interstitial solid solutions (I) to 1 for substitutional solid solutions (S), with R e = ½ for disubstitutionals (S2, split interstitials).
R A Y et a l . :
SUPERCONDUCTIVITY
OF N O N S U B S T I T U T I O N A L
T h e d e p e n d e n c e o f C~ on t h e solute c o n c e n t r a t i o n x is t h e n given b y Ce~ = Z + (z - - R o Z ) "
x 1 +
(Re - - 1)x
(6)
where Z, z are t h e valences of s o l v e n t a n d solute, respectively. F o r P b ( A g ) a n d P b ( A u ) a t s m a l l solute contents, Z -----4, z = 1, x ~ 1 a n d (6) r e d u c e s t o : C~ ~-~ 4 + x
for i n t e r s t i t i a l s (I),
(7)
C~ ~ 4 - - x
for d i s u b s t i t u t i o n a l s ($2).
(8)
C~z in (8) is i d e n t i c a l to C~z for a s u b s t i t u t i o n a l P b solid solution w i t h a t r i v a l e n t solute, where R~ = 1, z ~ 3. To t h e e x t e n t t h a t C~ d e t e r m i n e s t h e second t e r m in (3) a n d hence ((~Tc)valence, t h e slope o f (~T~ for d i s u b s t i t u t i o n a l P b ( A g ) s h o u l d be n e g a t i v e , as it is for P b ( I n ) , a n d Pb(Cd) [Fig. l ( a ) ] a n d c o m p a r a b l e in m a g n i t u d e to t h a t of P b ( I n ) , as observed. F o r P b ( A u ) [Fig. l(b)], t h e slope is also n e g a t i v e , b u t its m a g n i t u d e far exceeds t h a t o f Pb(T1) (see below). F o r b o t h P b ( A g ) a n d P b ( A u ) , T~ does n o t increase as w o u l d be e x p e c t e d f r o m (7) for i n t e r s t i t i a l solid solutions. W e conclude t h e r e f o r e t h a t R~ ~ ½ for t h e s e alloys. T h e c o n t r i b u t i o n o f t h e i m p u r i t y frequencies t h r o u g h t h e first t e r m of (3) is discussed n e x t . I n P b alloys w i t h solute valences b e t w e e n 3 a n d 5, t h e slope of 5To is a l w a y s larger for t h e l i g h t e r o f t w o i s o v a l e n t solutes. (11) P b ( A g ) a n d P b ( A u ) follow t h i s rule [Fig. 1(a)]; t h e i n t e r a t o m i c force c o n s t a n t s K~ can be a s s u m e d t o be n e a r l y e q u a l for b o t h solutes a n d t h e frequencies of t h e i m p u r i t y m o d e s are t h e n a p p r o x i m a t e d b y t h e m a s s r a t i o s ( M i m p / M p b ) -1/2. H o w e v e r , t h e site g e o m e t r y suggests t h a t for n o n s u b s t i t u t i o n a l solutes such as A g or Au, K i increases over t h e v a l u e s in s u b s t i t u t i o n a l solid solutions. This is confirmed b y M o s s b a u e r r e s o n a n c e m e a s u r e m e n t s of t h e recoil-free f r a c t i o n f : for In(Co), where Co is a s s u m e d to be i n t e r s t i t i a l , (16) f is f o u n d t o be t h r e e to four t i m e s larger t h a n t h e values e s t i m a t e d for s u b s t i t u t i o n a l Co ; for Au(Co), t h e r e is a n increase o f f on passing f r o m a s u b s t i t u t i o n a l t o a dis u b s t i t u t i o n a l solid s o l u t i o n which c o r r e s p o n d s t o a decrease of t h e m e a n - s q u a r e d solute a t o m d i s p l a c e m e n t
SOLID SOLUTIONS
1337
b y 35 p e r cent. (17) T h e increase o f K i p r e s e n t in b o t h s o l u t i o n t y p e s will r e s u l t in localized h i g h - f r e q u e n c y m o d e s which raise T c A c o r r e s p o n d i n g , s u b s t a n t i a l increase o f Tc for i n t e r s t i t i a l solutes is d e r i v e d in A p p e l ' s analysis. (14) F r o m t h e s e c o n s i d e r a t i o n s we conclude t h e n t h a t t h e o b s e r v e d decrease o f T~ in P b ( A g ) a n d P b ( A u ) is n o t d u e to t h e f r e q u e n c y t e r m o f (3). I t m u s t t h e r e f o r e be d u e to t h e v a l e n c e effect t e r m t r e a t e d a b o v e which led to R~ ~ ½. This result agrees w i t h r e c e n t conclusions f r o m i n d i r e c t evidence (ls~ for t h e f o r m a t i o n o f A u d i - s u b s t i t u t i o n a l s (considered as d i - i n t e r s t i t i a l - v a c a n c y clusters, I 2 V ) in e q u i l i b r a t e d P b ( A u ) [from r e s i s t a n c e v a r i a t i o n s a n d p r e c i p i t a t i o n r a t e s of A u P b a in P b ( A u ) ] . I t w o u l d be o f i n t e r e s t t o s t u d y T c in o t h e r nons u b s t i t u t i o n a l m e t a l - m e t a l solid solutions, especially t h o s e w i t h R~--~ 0, w h e r e T c m a y increase s t r o n g l y on alloying. ACKNOWLEDGEMENT
S u p p o r t of t h i s w o r k b y t h e Office o f N a v a l R e s e a r c h u n d e r C o n t r a c t N14-68-A-207-3 is g r a t e f u l l y acknowledged. REFERENCES
1. W. SEITH and A. KEIL, Z. phys. 22, 350 (1933). 2. B. F. DYSON, T. R. ANTHONY and D. T~RNRULL, J. appl. Phys. 37, 2370 (1966). 3. T. R. ANTHONY, in Vacancies and Interstitials in Metals, edited by A. SEEGER. Wiley (1970). 4. B. C. GIESSEN and R. H. WILLENS, in Phase Diagrams, edited by A. M. ALPER, Vol. III, p. 103. Academic Press (1970). 5. B. C. GIESSE~, P~. RAY and S. i . HAE~, Phys. Rev. Lett. 26, 509 (1971). 6. R. RAY, M. SO~INI and B. C. GIESSEN, Solid State Commun. 10, 163 (1972). 7. C. BORROMEE--GAuTIER,B. C. GIESSEN and N. J. GRANT, J. chem. Phys. 48, 1905 (1968). 8. R. RAY and B. C. GIESSE:N,to be published. 9. C. C. TSUEI and L. R. NEWKIRK, Phys. Rev. 183, 619 (1969). 10. L. R. NEWKIR~ and C. C. TSUEI, Phys. Rev. B3,755 (1971). 11. E. NEMBACH,J. Phys. Chem. Solids 29, 1205 {1968). 12. D. M. GINSBERG, Phys. Rev. 138, A1409 (1965). 13. D. MARXOWlTZand L. P. KADA~OFF,Phys. Rev. 131, 563 (1963). 14. J. APPEL, Phys. Rev. 153, 421 (1967). 15. J. R. ANDERSONand A. V. GOLD, Phys. Rev. 139, A1459 (1965). 16. P. A. FLIN~, U. GO~SE~, R. W. GRANT and R. M. I-IOUSELY, Phys. Rev. 157, 538 (1967). 17. C. F. STEEX, D. G. HOWARD and R. H. NUSSBAU~, Solid State Commun 9, 865 (1971). 18. A. ROSSOI~IMOand D. TUR~BULL, Acta Met. in press.