BOND EQUILIBRIUM THEORY IN COVALENTLY BONDED ALLOYS

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BOND EQUILIBRIUM THEORY IN COVALENTLY BONDED ALLOYS

M. Cutler

To cite this version:

M. Cutler. BOND EQUILIBRIUM THEORY IN COVALENTLY BONDED ALLOYS. Journal de

Physique Colloques, 1981, 42 (C4), pp.C4-201-C4-204. �10.1051/jphyscol:1981441�. �jpa-00220898�

JOURNAL DE PHYSIQUE

CoZZoque C4, suppZ6ment au nO1O, Tome 42, octobre 1981

BOND E Q U I L I B R I U M THEORY I N C O V A L E N T L Y BONDED A L L O Y S

M . C u t l e r

Physics Department, Oregon S t a t e University, CorvaZZis 97330 Ore., U.S.A.

Abstract.- Bond e q u i l i b r i u m t h e o r y (BET) has been extended t o i n c l u d e polymer chains whose u n i t s are molecular groups attached t o o t h e r u n i t s by one, two, o r t h r e e l i n k s . Because As-Se and s i m i l a r a l l o y s c o n t a i n h i g h d e n s i t i e s o f atoms w i t h t h r e e f o l d (3F) bonding, i t i s l i k e l y t h a t a l a r g e f r a c t i o n o f t h e As atons o f t h e l i q u i d a l l o y a r e i n molecular groups. I n i t i a l r e s u l t s o f a study o f AsxSel-x a r e presented.

Bond E q u i l i b r i u m Theory (BET) f o r Atomic Constituents.- I n a study o f Se-Te a l l o y s (11, i t has been shown t h a t t h e c o n c e n t r a t i o n o f a bond d e f e c t y c o n t a i n i n q n bonds on.an atom o f chemical species A i s

where q i s t h e charge number, 6 = l / k T , and EF i s t h e Fermi energy. (Concentra- t i o n s are normalized t o the d e n s i t y o f atoms i n t h e a l l o y . )

Y

g i s the f r e e energy

Y

o f f o r m a t i o n from the f u l l y bonded atom. Aside from t h e polymer f a c t o r pn, t h i s expression i s t h e same as f o r the e q u i l i b r i u m c o n c e n t r a t i o n of c r y s t a l i m p u r i t i e s o r l a t t i c e d e f e c t s i n t h e t h e o r y o f KrBger and Vink (2). XA i s t h e f u g a c i t y o f an A atom. I t s value i s determined by t h e requirement t h a t t h e t o t a l c o n c e n t r a t i o n o f the A atoms adds up t o t h e atomic f r a c t i o n x A i n t h e a l l o y . I f bondinn i s nonran- dom, t h e expressions a r e somewhat more complicated. The d i s c u s s i o n i n t h i s paper w i l l be r e s t r i c t e d t o random bonding, i . e . , t h e energy o f an AB bond i s assumed equal t o t h e average f o r AA and BB.

The polymer f a c t o r s pn a r e t h e r e s u l t o f t h e t o p o l o g i c a l c o n s t r a i n t s which r e - l a t e t h e number o f p o s s i b l e c o n f i g u r a t i o n s o f polymer molecules t o t h e t o t a l con- c e n t r a t i o n s cn o f c o v a l e n t l y bonded atons which have n bonds, f o r t h e d i f f e r e n t values o f n. The t h e o r y has been worked o u t f o r polymers formed from atoms which have 1, 2, o r 3 bonds, i n an aporoximation which ignores s t r u c t u r e s c o n t a i n i n a loops. Since each 3F atom creates a c h a i n branch which must be terminated by a 1F atom, t h e c o n c e n t r a t i o n o f polymer molecules i s (cl-c3)/2. The branch r a t i o

X

= 2c3/(c -c 1 3) , i.e., t h e average number o f c h a i r branches per molecule, i s an im- p o r t a n t parameter i n t h e theory. The polymer f a c t o r s have values given by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981441

C4-202 JOURNAL DE PHYSIQUE

B and h a r e parameters r e l a t i n q t o t h e c o o r d i n a t i o n and bond angle r e s t r i c t i o n s as described i n Ref. 1. Appropriate values f o r chalcogenide a l l o y s seem t o be B = 6 and h = 2. b i s a parameter whose value increases s l o w l y w i t h A, fr

? m

b = 1 f o r A 3 t o 2.618 when A + - (1). I n t h e l i m i t A >> 1, pl =A and p 3 ~ A -

.

The values o f t h e f r e e energy parameters a r e o f t e n unknown, and one hopes t o d e r i v e t h e i r values by comparison o f t h e p r e d i c t i o n s o f BET theory w i t h experimen- t a l data. I t i s t h e r e f o r e d e s i r a b l e , i f p o s s i b l e , t o use t h e BET equations t o pro- v i d e r e l a t i o n s between concentrations o f important c o n s t i t u e n t s , w i t h o u t knowing t h e values o f t h e parameters gv o r a t most having a rough estimate. I n t h i s ap- proach, a v a l u a b l e consicieratidn i s the e f f e c t o f A on t h e r e l a t i v e concentrations o f I F and 3F c o n s t i t u e n t s . I f c,-c3 ⁺ 0, A becomes l a r g e , causing pl t o increase and p3 t o decrease, which ensures t h a t cl/cg>l.

BET f o r l l o l e c u l a r Constituents.- The d r i v i n a mechanism which tends t o increase t h e r a t i o c1/c3 i s o f course t h e increase i n entropy due t o t h e l a r g e r numbers o f smaller molecules. I n Se-Te a l l o y s , r i n g molecules attached t o chains by a s i n g l e 3F atom have been found t o p l a y an important r o l e as a r e s u l t o f t h i s mechanism (3).

Such r i n g s a c t as 1F c o n s t i t u e n t s which a r e molecular r a t h e r than atomic, and an ex- t e n s i o n o f EET t o i n c l u d e 1F molecular c o n s t i t u e n t s has been d e r i v e d i n Ref. 3. A l - though t h e d i s c u s s i o n was i n terms o f r i n g molecules, t h e b a s i c t h e o r y i s a p p l i c a b l e t o any molecular c l u s t e r which i s attached t o a chain by a s i n g l e l i n k . I n t h e case o f Se-Te, t h e I F r i n s become important because t h e thermal e x c i t a t i o n o f i o n p a i r s _{( I F}

_D-

i o n s and 3F D

9 .

i o n s ) r e q u i r e s l e s s energy than e x c i t a t i o n o f a p a i r o f neu- t r a l danql i n q bonds (1 F D* atoms). S i n g l y attached

' R

r i n g s c o n t a i n i n g a

D+

atom p r o v i d e a means o f forming i o n p a i r s (D-

-

R') which are b o t h 1 F c o n s t i t u e n t s .

As-Se and r e l a t e d a l l o y s have h i g h d e n s i t i e s o f 3F atoms so t h a t c3 i s l a r g e The r e s u l t s i n Se-Te suaqest t h a t molecular qroups p r o v i d e a mechanism f o r incorpo- r a t i n g t h e 3F atoms o f As-Se i n molecular s t r u c t u r e s i n which c3 < cl. I n t h i s des- c r i p t i o n , cl, c2, o r c3 r e f e r t o concentrations o f molecular c o n s t i t u e n t s w i t h 1, 2, o r 3 l i n k s t o o t h e r members o f a polymer chain. I t has been p o s s i b l e t o g e n e r a l i z e t h e t h e o r y presented i n Ref. 3 t o i n c l u d e molecular y o u p s which have I F , 2F, 3F bonding. The o r i g i n a l work a l s o takes i n t o account molecular groups which are n o t attached t o chains, and t h e r e f o r e can be catagorized as OF c o n s t i t u e n t s . The r e - s u l t s a r e described as f o l l o w s : The an n-F molecular c o n s t i t - uent based on a molecule which i s composed o f v,...

n v -1

cmy = pn Om pm a ex~[-f3(gmy ⁺ qy E F ) l ( 5 ) i s t h e f r e e energy o f f o r m a t i o n o f t h e molecule w i t h a s t r u c t u r e m from t h e atdms i n t h e i r r e f e r e n c e s t a t e , and q i s t h e charge number. Pm(< 1 ) i s t h e r a t i o

Y

o f t h e number o f c o n f i g u r a t i o n s o f t h e molecule t o t h e number o f c o n f i g u r a t i o n s o f v, atoms i n a u n r e s t r i c t e d l i n e a r chain. ( I t i s t h e same as Rm i n Ref. 3.)

0'; i s a m u l t i p l i c i t y f a c t o r d e s c r i b i n g d i s t i n g u i s h a b l e c o n f i g u r a t i o n s r e s u l t i n g from p l a c i n g the l i n k s a t d i f f e r e n t places i n t h e molecule, as can occur, f o r i n - stance, i n 2F r i n g s . I n t h e case o f a b i n a r y a l l o y c o n t a i n i n g A and B atoms w i t h f u g a c i t i e s XA and XB, t h e expression f o r cn f o r a molecular c o n s t i t u e n t c o n t a i n i n g

my V~

v A A atoms and vR

- E

atoms would a l s o i n c l u d e a f a c t o r XA ^{. .} XB ^-

.

X, i s a parameter which appears i n t h e t h e o r y o f Refs. 1 and 3, and which be- -

comes important i n s i t u a t i o n s where molecular c o n s t i t u e n t s c o n t a i n i n ? many atoms need t o be considered. I t s value can be e x ~ r e s s e d by

xa

= 2c2/ (cl +2c2+3c3) v -1 ( 6 )

When c2 >> clfc3, Xa i s s l i g h t l y l e s s than 1. But t h e f a c t o r Xa i n Eq. 5 causes

'I,

the d i s t r i b u t i o n o f s i z e s o f molecular groups t o f a l l o f f s h a r p l y f o r vm > (1-Xa)-'.

In extending the theory to molecular constituents, each molecular group is treated as single particle as far as the topology of the polymer molecules is con- cerned. Thus all the atoms in OF molecules and all but one of the atoms of each attached molecular group are excluded. This effect is expressed in terms of the fraction

@

of "effective" polymer units:

0

1

= C1+C2C3 =

1 - c " ^{- - 1 +c2 +c3}

⁾

. ⁽⁷⁾

m m m.v my

MY

my

Application to AsxSel-x Alloys.- A start has beeinmade in an investigation of the behavior of AsxSel-x alloys in terms of BET. It seems likely that alloys at compo- sitions near As2Se3 contain complicated molecular structures as exemplified by the molecule As4Se6, which is known to occur in the gaseous phase. It has a cage-like structure containing four rings. A wide variety of analogous multiply-connected structures would need to be considered if large molecular groups are important. As discussed in Ref. 3 for the case of rinqs, steric constraints limit multiply-con- nected structures to a minimum number of

% 8

atoms in a ring. Consequently it is a matter of great interest to learn whether Xa is small enough in As-rich alloys to restrict the size of the important molecular groups to the smallest ones consistent with the constraints of the bond angles.

The approach which has been taken is to first study simplified models in the limit of small x (2 O.l), where only ring constituents need to be considered. A useful first step is to neglect thermally excited bond defects, since their con- centrations are apt to be small compared to those of IF, 2F, and 3F molecular con- stituents formed from normally bonded As and Se atoms (at least at low T).

In Fig.

1 ,

the dependence of

A,

a , and the concentrations of constituents on alloy composition is presented for a model in which only OF and 1F ring constit- uents are considered, in addition to 2F Se atoms and 3F As atoms. The parameters determining the ring distributions are the same as in Ref. 3. Each 1F ring has one Se atom replaced by an As atom, so that c1+c3

= X.

As seen in Fig. 1, cl is

slightly larger than x/2 and c3 is slightly smaller, the differences decreasing with x due to the increase in

A

(also plotted). The concentration of free rings co be- comes small compared to cl as x increases since it does not depend on pl, which in- creases with x.

The most significant result in this model is the behavior of the distribution of ring sizes. The concentration of an n-atom ring is proportional to (XSXa) n-1 .

Since XaXS

=

c2, the density of n-atom molecular groups falls off sharply when n ? 1 - c ) . Fig. 1 includes plots of c2 and the average number of atoms in a ring

<n>. It is seen that c2 becomes small-enough at moderate As concentrations so that

<n> approaches the minimum value

8

set by the model.

This model describes a non-thermal liquid. One expects that thermally excited IF or 3F bond defects will modify the behavior when their concentrations become comparable to c1 -c3.

Conclusions.- The groundwork has been laid for an application of BET to the molecu-

lar structure of alloys of the type As-Se. The alloy is viewed as simply-connected

polymer chains composed of IF, 2F, and 3F constituents which may be molecular or

atomic. Preliminary studies support the conclusion that only molecular groups of

the minimum size consistent with steric constraints are important. At low As con-

centrations considered so far, these would be rings with

% 8

atoms. The premises

of the theory require that c,

>

c3. Therefore when the ratio of As/Se atoms be-

comes larger, rings containing three or more As atoms will become important, and

molecular groups would have to be formed with more complicated multiply-connected

JOURNAL DE PHYSIQUE

I0O0

s t r u c t u r e s . The conclusions are c o n s i s t e n t w i t h some o f t h e concepts about t h e s t r u c t u r e o f As-Se 10-3-

a1 l o y s d e r i v e d from experimental i n f o r m a t i o n ( 4 ) . They a r e a l s o t o be compared w i t h t h e models CO

developed by P h i l l i p s f o r amorphous chalcogenide a l l o y s ( 5 ) . Our models agree i n t h e sense t h a t mo- l e c u l a r groups a r e taken t o be t h e b a s i c s t r u c t u r e u n i t s , b u t the BET t h e o r y leads t o an emphasis on chai n-1 i ke s t r u c t u r e s composed o f small molecular ,o-5

groups r a t h e r than l a y e r - l i k e s t r u c t u r e s .

The present study i s based on models which assume random bonding. Spectroscopic and d i f f r a c - t i o n s t u d i e s i n d i c a t e a s t r o n g tendency f o r a l t e r -

10-61 !;/

nate bonding i n amorphous As-Se a l l o y s near t h e _{lo l4 10-2}

compositions As2Se3, which suggests non-random _{~t % AS}

bonding (4). However t h e d i f f e r e n c e i n e l e c t r o - n e g a t i v i t i e s o f As and Se i s small and an e m p i r i - c a l formula due t o P a u l i n g ( 6 ) suggests t h a t a mixed bond has a s t a b i l i z a t i o n energy o f o n l y 0.16

eV. Since kT % - 0 5 eV a t the glass t r a n s i t i o n o f Fig. 1 Results o f a s i m p l i f i e d As2Se3, t h i s would g i v e an appreciable concent'ra- BET model f o r AsxSelmx.

t i o n o f l i k e bonds. I t seems p o s s i b l e t h a t t h e multiple-connected s t r u c t u r e s o f minimum s i z e ,

l i k e t h e As4Se6 cage molecule, r e q u i r e a l t e r n a t e bonding because o f s t e r i c con- s t r a i n t s . Therefore t h e observed a l t e r n a t e bonding, may be t h e i n d i r e c t r e s u l t of s t e r i c c o n s t r a i n t s i n c o n j u n c t i o n w i t h t h e l i m i t a t i o n t o small molecular u n i t s . This work has been supported by the N a t i o n a l Science Foundation w i t h g r a n t s DMR 77-19035 and DMR 80-23682.

(1 ) CUTLER M., Phys. Rev. B20 (1979) 2981

.

(2) K R ~ G E R F.A., "The Chemistry o f Imperfect C r y s t a l s ,I1 North Holland Amsterdam (1 964).

(3) CUTLER

PI.,

BEZ W.G., Phys. Rev., t o be published.

(4) LUCOVSKY G.

,

HAYES T.M., i n "Amorphous Semiconductors

,"

Ed. M.H. Brodsky

,

Springer Verlag, B e r l i n (1979) 21 5.

(5) PHILLIPS J.C., J . N o n - C r y s t a l l i n e S o l i d s 3 4 (1,979) 153.

(6) PAULING L., "The Nature of t h e Chemical Bond, 3rd. Ed., Cornell Univ. Press, I t h a c a (1960).