EFFECT OF

OF

ORDERED

VACANCIES

ON

THE

RESISTIVITY

ALLOYS J

Z.

A.

Matysina

and

E. A.

UDC 669.017:539.21

Matysina

An analysis is made of the effect of lattice vacancies in a binary o r d e r e d substitutional a l loy with a bce lattice on the residual resistivity. A calculation shows that the c o n c e n t r a tion dependence of the r e s i s t i v i t y loses its s y m m e t r i c form. At c e r t a i n compositions vacancies can r e d u c e the residual r e s i s t i v i t y , and at other compositions they can i n c r e a s e it. An experimental study of the a s y m m e t r y of the curve will yield information about the atomic interaction e n e r g i e s in an alloy. The r e s i s t i v i t y is e x t r e m e l y sensitive to various types of s t r u c t u r a l defects in alloys, which f r e quently determine the alloy p r o p e r t i e s which a r e of practical importance, e.g., the strength. F o r this r e a s o n and in connection with the effort to find alloys with prespecified p r o p e r t i e s , it is worthwhile to study the resistivity. One type of s t r u c t u r a l defect in an alloy consists of a vacancy at a lattice point. When alloys a r e annealed at t e m p e r a t u r e s slightly below the melting point, a c e r t a i n number of v a c a n c i e s b e c o m e e s t a b lished, in a concentration which may r e a c h ~1%. Vacancies in a metallic compound lead to an additional r e s i s t i v i t y ; for alloys quenched to low t e m p e r a t u r e s after a prolonged h i g h - t e m p e r a t u r e annealing, a residual r e s i s t i v i t y is found. A t h e o r y of this r e s i d u a l r e s i s t i v i t y due to v a c a n c i e s was derived in [1] (see also [2]) for the c a s e of d i s o r d e r e d alloys. In intermetallic compounds, however, and in c e r t a i n metallic compounds, e.g., Ni~A1, a high degree of o r d e r is known to be p r e s e r v e d right up to the melting point. The o r d e r - d i s o r d e r transition t e m p e r a t u r e for a binary alloy is given by Z~

Tk = - - c~ c2,

(1)

w h e r e c 1 and c 2 a r e the a t o m i c concentrations of the alloy components; w is the o r d e r i n g energy; z is the coordination number; and k is the Boltzmann constant. The quantity U 0 = zw gives the i n c r e m e n t in the internal energy of the completely o r d e r e d alloy caused by the transition of one atom f r o m ~its own ~ lattice point to na different one. n If U 0 is large, as, e.g., for Ni3Al , the alloy r e m a i n s o r d e r e d up to quite high t e m p e r a t u r e s , at which the vacancy concentration is quite high, and the r e s i d u a l r e s i s t i v i t y due to v a c a n c i e s can be determined experimentally. We r e p o r t here a study of the r e s i d u a l r e s i s t i v i t y of o r d e r e d alloys due to vacancies. We consider a b i n a r y o r d e r e d substitutional alloy with a bcc lattice formed f r o m NA a t o m s of species A and NB a t o m s of species B (NA + NB = 2N). The atomic concentrations of the alloy components a r e NA

cA=~,

cB-

NB 2N

.

(2)

We denote by VAA, VBB, and VAB (the negative values of) the interaction energies of neighboring pairs of atoms. We a s s u m e that the differences among these energies a r e small in c o m p a r i s o n with kT, where T is the annealing t e m p e r a t u r e . We denote by N(~), N~),~ and nt, respectively, the n u m b e r s of a t o m s U k r a i n i a n - R u s s i a n Union T e r c e n t e n a r y State University, Dnepropetrovsk. T r a n s l a t e d f r o m I z v e s t i y a Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 81-87, September, 1970. Original a r t i c l e submitted July 8, 1969. 9 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. ,4 copy of this article is available from the publisher for $15.00.

1187

o f species A and B and of vacancies at lattice points of the f i r s t kind; and we denote by tgN~), N~),t~ and n2 the s a m e quantities for lattice points of the second kind. These quantities a r e related by

N~

)+hi =n,

N ~ ' + N~ ) + n . . , = n ,

(3)

where 2rt

:

/4)

2 N + ni + n,,_.

Statistical theory [3] and the a v e r a g e - e n e r g y method yield the following e x p r e s s i o n s for n 1 and n~:

V n2 = N V

cB--1/'2~q exp 4(VAB--V~8):~ exp --u~

(s)

CA - - 1/2 N exp CA + 1/2~

4 (v,4~ "OAA) toT -

-

7~

exp -- u----3-~ uT

where ~? is the d e g r e e of l o n g - r a n g e o r d e r , given for this case by* ~=

2

--CA

,

(6)

N

the quantity tt~ = ao + w~'-'

(7)

as the v a c a n c y - f o r m a t i o n energy, and uo ~ 4 (C~VAA + C~ VBB + 2CACB~)AB) = 4 [VBB + 2 (VAB - - VBB) CA - - WC~], w = 2VAB - - VAA - - ~'~B.

(8) (9)

Equations (5) show that in an o r d e r e d alloy vacancies a r e redistributed among points, and the number of vacancies at lattice points of a given kind is not the same as in the d i s o r d e r e d alloy. As will become c l e a r below, this c i r c u m s t a n c e has a c e r t a i n effect on the concentration dependence of the residual r e sistivity. We will now c a r r y out a qualitative analysis of the residual resistivity, making the following simplifying a s s u m p t i o n s : 1) We a s s u m e that the potential energies VA and V B of an electron in the fields of ions A and B a r e approximately equal, i.e., that the difference VA - V B is small. 2) We r e s t r i c t the analysis to the approximation of n e a r l y f r e e electrons, in which case potential energies VA and VB a r e small in c o m p a r i s o n with the a v e r a g e e l e c t r o n kinetic energy. In using these assumptions, we will r e s t r i c t the calculations to quantities of only a second o r d e r of smallness. The potential e n e r g y of an e l e c t r o n in the field of the lattice ions is 11

i~

(10) s=l

x=l

where V s z is the potential e n e r g y of an electron in the field produced by lattice point x of e l e m e n t a r y cell s ; r is the v e c t o r coordinate of an electron; R s ~ is the v e c t o r leading f r o m the initial coordinate to the l a t tice point ~ of e l e m e n t a r y cell s; and ~ is the n u m b e r of lattice points p e r e l e m e n t a r y cell. We t r e a t the vacancies as the "atoms" of a third component; the potential energy of an e l e c t r o n in the field of a v a c a n c y is VD = 0. As usual, we adopt as a z e r o t h approximation a completely o r d e r e d alloy c o n sisting of effective ions. In the zeroth approximation the e l e c t r o n potential e n e r g y is thus n

(11) S=I

x~l

* Here ~? is defined in such a manner that it is equal to the degree of l o n g - r a n g e o r d e r of a binary AB alloy containing no vacancies.

1188

where

(12)

PA VAAvPB VB-F P(~) Vo, r e s p e c t i v e l y , for lattice points of the f i r s t and second kinds. In Eqs. (12), p~) is the probability that a l a t tice of kind i (i = 1, 2) is filled by an a t o m of s p e c i e s a (a = A, B, D). F o r an alloy with v a c a n c i e s we d e fine these probabilities to be p(0= N(~'),

pg)= n,_;

1l

(13)

n

the following e x p r e s s i o n s w e r e derived f o r t h e s e probabilities in [3]:

p(~)=(CA+ I/2"q)(I q-CD) -I, p(~)=(cA--t/23)(1-FCD) -1, p(~) = (CB -- 1/2~-- ~)(1 -F CD)-1, p(~) = (CB'-F 1/2~3-l-~) (1 -FCo) -~, n,

(14)

(1 +co)-', p~)=~--(l§

where CD ~

nl~ + n2

(15)

2N

is the a t o m i c c o n c e n t r a t i o n of v a c a n c i e s in the alloy, and =

nl -- n2

(16)

2N Calculation of the s q u a r e modulus of the m a t r i x e l e m e n t of the p e r t u r b i n g energy, defined by V' = V - V0, yields 2

Xi

I V',k.12 = r t ~ ~ (p~)p~'lV;~,. -- V~,,.]"q-... ~,c, t a,k'l i=l xi=l 2

~-p~) p~) ]V;i.~,, 12) = 2n .','~ O,p~) p(~' AAs+ .;p~) pg) Aa- ~ .~i p~) p~> AB),

(17)

kk' = Vokk' = f e ~qr V~ (r) ttk (r) ul. (r) d2.

(18)

where

H e r e uk(r) is the periodic p a r t of the wave function of an e l e c t r o n having wave v e c t o r k; q = k - k'; ~ is the c r y s t a l volume; and vi = ki//x is the concentration of lattice points of kind i. Substituting p r o b a b i l i t i e s (14) into Eq. (17), we find the dependence of the s q u a r e m a t r i x e l e m e n t [V~k,[2 , which is p r o p o r t i o n a l to the r e s i d u a l r e s i s t i v i t y of the alloy, on the alloy composition, d e g r e e of l o n g - r a n g e o r d e r , v a c a n c y c o n centration, and the quantity 6, defined by Eq. (16). This calculation yields the following for the r e s i d u a l resistivity:

w h e r e At, A2, and A 3 a r e positive constants independent of the c o m p o s i t i o n and d e g r e e of l o n g - r a n g e o r d e r . In deriving Eq. (19) we a s s u m e d n i / N , n2/N , CD, and 8 to be s m a l l , r e t a i n i n g only e x p r e s s i o n s linear in t h e s e quantities. Equation (19) can be conveniently r e w r i t t e n as

1

[A2

_kA3c2_2A,(clc.2

1 .~

(20)

1189

T h e f i r s t t e r m in this e q u a t i o n , Pl : Al(clc2 - ~ 2 / 4 ) , i s the r e s i s t i v i t y of a b i n a r y o r d e r e d a l l o y c o n t a i n i n g no v a c a n c i e s ; t h i s r e s i s t i v i t y i s d e s c r i b e d by a c u r v e w h i c h i s s y m m e t r i c a b o u t the p o i n t s c t = 0.5. F o r a n a l l o y h a v i n g a m a x i m u m d e g r e e of l o n g - r a n g e o r d e r ,

1

2cl

"qm :

for

c ~ l (21)

t 2 C2 for t

c1 ~ ~ , 2

t h i s c u r v e i s t h e d a s h e d c u r v e in F i g . 1. T h e s e c o n d t e r m in Eq. (20), P2 = cD(Aacl + A 3 c 2 ) = PACl + PBC2, w h e r e PA and PB a r e the r e s i s t i v i t i e s of p u r e m e t a l s A and B c o n t a i n i n g v a c a n c i e s , i s t h e r e s i s t i v i t y of the d i s o r d e r e d a l l o y due to t h e p r e s e n c e of h o l e s in t h e p u r e m e t a l s A and B, a v e r a g e d o v e r the c o n c e n t r a t i o n . T h i s t e r m would g i v e a l i n e a r d e p e n d e n c e on c, w i t h c D = c o n s t . It w a s shown in [3] that the c o m p o s i t i o n d e p e n d e n c e s of n I and n 2 d i s p l a y m i n i m a n e a r t h e s t o i e h i o m e t r i c c o m p o s i t i o n (with VBB > vAA, t h i s m i n i m u m l i e s on the l e f t of p o i n t c 1 = 0.5, w h i l e w i t h VBB < VAA i t l i e s on t h e r i g h t ) . A c c o r d i n g l y , the t o t a l n u m b e r of v a c a n c i e s in the a l l o y (or the q u a n t i t y CD) c h a r a c t e r i s t i c a l l y f a i l s off t o w a r d the s t o i c h i o m e t r i c c o m p o s i t i o n . F o r s t o i e h i e m e t r i c a l l o y s w i t h a m a x i m u m d e g r e e of o r d e r t h i s m i n i m u m i s z e r o . T h e r e s i s t i v i t y due to the s e c o n d t e r m , PAC94 + PBC2, i s s h o w n by the s o l i d c u r v e in F i g . 2 f o r the c a s e ~? = ~ m and f o r a s t o i e h i o m e t r i c a l l o y . H e r e t h e m i n i m u m is z e r o and o c c u r s a t c 1 = 0.5. In t h e c a s e 77 ~ ~?m, the m i n i m u m i s n o t z e r o , and i t l i e s a t c 1 = 0.5 if VAA = VBB. T h e t h i r d t e r m i n Eq. (20), P3 = - 2 A 1 ( c l c 2 ' ~ 2 / 4 ) , l o w e r s the d a s h e d c u r v e in F i g . 1 s l i g h t l y , s i n c e i t i s n e g a t i v e ; t h e s y m m e t r y of the c u r v e i s l o s t b e c a u s e of the a s y m m e t r i c d e p e n d e n c e of CD on c 1. T h e f o u r t h t e r m in Eq. (20), c a n c a u s e the d a s h e d c u r v e in F i g . s i n c e 6 c h a n g e s s i g n a t s o m e point d e g r e e of o r d e r ~ m , g i v e n b y (21),

n,

[~

/-ca

_

P4 = - 5 ~ (A 1 - A 2 + A a ) / 2 , w h i c h v a n i s h e s e x c e p t f o r o r d e r e d a l l o y s , 1 to b e c o m e m u c h m o r e a s y m m e t r i c t h a n do t h e s e c o n d and t h i r d t e r m s , n e a r the s t o i c h i o m e t r i c c o m p o s i t i o n . F o r e x a m p l e , f o r the m a x i m u m we find

-

ca

-

8 ( v A . - v . ~ ) c,

exp

O for

C~

0 for

c~ < - -

g-c,--c.,

I

2

n2

N

exp

cx~-

2

2 exp

--un~ toT

toT

for c , ~ 1 , 2

4[vsB +2(vA~--v~8)cl] for c ~ < + 1 c , = -Z

4[VAA+2(VAS--VAA) Cj

for

C,~-, (24)

I

nT

0 for c~ = -

c, < L

2

1 2

] _ _ l / c , - - c ~ exp --4va-------~afor c l ) 1 . ~, toT ~ 2 F i g u r e 3a and b, s h o w s the c o m p o s i t i o n d e p e n d e n c e s of n 1 and n~, r e s p e c t i v e l y , f o r t h e m a x i m u m d e g r e e of o r d e r . In t h e c o m p l e t e l y o r d e r e d a l l o y w i t h c 1 -< 1 / 2 , v a c a n c i e s a r e c o n c e n t r a t e d a t l a t t i c e p o i n t s of the f i r s t kind, w h i l e with c I _> 1 / 2 t h e y a r e found a t l a t t i c e p o i n t s of t h e s e c o n d kind. T h i s

1190

(22)

(23)

8(vas--vaa)c2

( ]/- C~ - - Cl exp - - 4vs_______._sBfor

,

~r

l

~4vA~ for k

gT

,

"

i

~m

exp

KT

J - - : V ..... o,5 _ o,

o

~

o,s._ el

o

Fig. 1

/

Fig. 2

g nl . _ . #tf~e

$

n ]H

0 , $ C~

4Y, ~t

"1 '1'1 V'l'/i ,

o

----~ c,

g,

,

Fig. 3

Fig. 4

situation is found because a t o m s of species A and B tend to occupy "their own" lattice points in both c a s e s at the maximum d e g r e e of order. Figure 4 shows the concentration dependence of P4 for the case A l - A 2 + A 3 > 0. The solid c u r v e in Fig. 1 shows the total r e s i s t i v i t y of an ordered alloy containing v a c a n c i e s for the m a x i m u m degree of l o n g - r a n g e order.* This figure r e v e a l s an interesting effect: a completely o r d e r e d s t o i c h i o m e t r i c alloy quenched after a prolonged and h i g h - t e m p e r a t u r e annealing must display (if not a vanishing resistivity) a r e s i s t i v i t y lower than those of the pure components of the alloy, treated in the s a m e manner. If the d e g r e e of l o n g - r a n g e o r d e r in the alloy is not maximal, the quantities n I and r~ will be nonvanishing over the entire concentration range. F o r the s t o i c h i o m e t r i c alloy we have ~/"~-~

n,=N

4(VAB-

V l~exp

n2 = N

/1

~

"OBB)~q--tt,~

toT

- - ~q

exp-

'

(25)

4 ( V A B - - V A A ) ~ - - U~,

gT

(26) tt~ = 4 v A 8 - - ( l - - ~q2) w ,

4~BB'~

=--exp

2

If VBB > VAA, we have exp

,+m

--4vaB~.~exp --4VAA'~ •T

e

~r

4~AA ~

-- e

~r

(27)

and 6 < 0.

KT

We find that 5 vanishes at some value c 1 < 1 / 2 . This case c o r r e s p o n d s to that discussed as an exa m p l e in [3]. In this case the effect of the fourth t e r m in Eq. (20) on the residual r e s i s t i v i t y is weaker on the interval [0, 1 / 2 ] than on the interval [1/2, 1]. By experimentally studying the concentration dependence of the residual r e s i s t i v i t y of alloys subjected to various kinds of heat t r e a t m e n t (alloys annealed at sufficiently low t e m p e r a t u r e s do not contain v a c a n c i e s , while alloys quenched after a prolonged annealing near the melting point contain a relatively large number of vacancies), we can evaluate the quantities Ap and Ap, (Fig. 1), which give, respectively, the i n c r e a s e and d e c r e a s e in the residual r e s i s t i v i t y at various compositions due to the p r e s e n c e of v a c a n cies. By knowing Ap and/Xp,, we can find some information about the a t o m i c interaction energies; in p a r ticular, we can d e t e r m i n e whether VAA or VBB is l a r g e r , i.e., whether the AA pair or the BB pair i n t e r acts m o r e strongly. * The effect of v a c a n c i e s on the r e s i s t i v i t y is exaggerated in this figure.

1191

LITERATURE Io 2. 3.

1192

CITED

A. A. Smirnov, Dokl. Akad. Nauk UkrSSR, No. 3, 172 (1953); No. 4, 250 (1954). A. A. Smirnov, Theory of the Resistivity of Alloys [in Russian], Izd. AN UkrSSR, Kiev (1960). A. A. Smirnov, Dokl. Akad. Nauk UkrSSR, No. 5 (1950).