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A DOUBLE HARD SPHERE MODEL FOR MOLTEN
SEMICONDUCTORS AND SEMIMETALS
B. Orton
To cite this version:
B. Orton.
A DOUBLE HARD SPHERE MODEL FOR MOLTEN SEMICONDUCTORS
AND SEMIMETALS. Journal de Physique Colloques,
1980,
41 (C8),
pp.C8-280-C8-283.
JOURNAL DE PHYSIQUE CoZzoqtre C 8 , suppZ&ment au n08, Tome 4 1 , aoiit 1980, page C8-280
A DOUBLE HARD SPHERE MODEL FOR MOLTEN SEMICONDUCTORS AND S E M I M E T A L S B.R. Orton
Physics Department, BruneZ University, Kingston Lane, Uxbridge, U.K.
Abstract.- The experimental interference functions for Si, Ge, Ga, Sn, Sb and Bi all exhibit a subsidiary maximum on the high K side of the first main peak. It is shown that this feature may be reproduced by a double hard sphere model which contains the following assumptions. (a) The liquid consists of two atomic species, A and B, corresponding to long and short distances of atomic separa- tion. (b) The interaction of species A and B gives the same atomic separation as between A species. Thus the total interference function of this binary mixture is a superposition of only two partial interference functions have been used for these partials. This model is discussed in the light of possible bonding in the liquid and the recent ideas of core polarization effects in these liquid metals. It is shown how this model may be extended to the binary alloy Cu-Sn.
1. Introduction.- It is now well established that diffraction measurements on the moltensemiconduc- tors, Si /1,2/ and Ge /1,2,3,4/ and the semimetals Ga /5,6,7/, Sn /2,8,10/, Sb /11,12,13,14/ and Bi /8,15,16/ give interference functions, I(K) (K= (47rsin@)/k, 28 = scattering angle,
X
= wavelength of incident radiation) which have main peaks which are broadened by a subsidiary maximum or shoulder on the high K side.The model proposed for liquid semiconductors 1181 and semimetals /19/, 1201, /21/ contains the £0110- wing features. (a) The atoms of these elements can come together to give either long A or short B dis- tances of atomic separation. ( b ) The interactomic separations between the A and B species are the same as between the A species. It has been shown 1181 that it is possible to divide the liquid up into partial interference functions IAA(K) and IBB(K) and express the total interference function as the superposition of only two partial functions ;
2
IT (K) = cA ( 1+cB)IAA(K) + cBIBB(K) (1
cA = atomic fraction of A component
2. Results.- The experimental data which were used in this work were chosen from the author's own mea- surements on Ge /3/ and Bi /15/, the work of Waseda for Si
111
and Sb 1131, and for Sn and Ga. North/a/
and Page et al. 151, respectively, provided the results.To match IT(K) given in Eq. (1) to the experimental I(K) the two partial interference functions IAA(K) and IBB(K) were provided by the Ashcroft and Lekner /17/ hard sphere I(Ku,q), where 0 was the
hard sphere diameter and the packing density.
The values of aA, oB, qA, '1 and cA were vat-ied B
in a systematic way until the best possible agree- ment was achieved. The values of these parameters are recorded in table 1.
I (K) and I (K) are shown in Figs. 1 and 2 particu-
exP T
lar note should be taken of the excellent agreement obtained over the whole of the main peak of IGa(K). Agreement for semiconductors was not so good. It was found possible to match the shape of the peak or height, for Si, but not both together. It may also be noted that the general agreement over the second and subsequent peaks was poor.
3. Discussion of the results.- From the evidence of the results considered above it is clear the the molten elements Si joins, Ge, Ga, Sn, Sb and Bi as describable in terms of a double hard sphere model. The nature of the atomic interactions which brings this about will now be discussed.
Table 1 gives the crystal structures of the elements mentioned in this work. They either show strong covalent bonds or some evidence of p type direc- tional bonding (Sn, Sb, Bi /23/), Ga shows certain evidence of Ga2 molecules in its structure /24/. It is the contention of the present work that the'di- rectional nature of these bonds is weakened in the liquid, but not completely lost, so the liquid metal is not wholly free electron like. The short inter- atomic distance comes about when the p type bond directions of a pair of adjacent atoms are correctly orientated. This means that if a pair of atoms bond together with short atomic separation, the remaining
surrounding atoms do not take up the short distance but stay at the longer distance because it is unli- kely that the orientation of the p type orbitals will be correct. The longer A separations would correspond to non-directional metallic bonds. Measu- rements which are sensitive to both s and p valence states are electron energy distribution curves obtained by x-ray or W photoemission.
Using UV radiation it has been found for Ga 125,261, Sn /27,28/ and Bi /28,29/ that s and p regions remain separated in the liquid state, similar to the results for the crystalline solid. Thus any bonding in the liquid could be expected to have the p type character of the solid. This double structure is in general agreement with various interatomic poten- tials that have been proposed and tested for Bi /30,31/ and Ga /32,33/.
4. Extension to Cu-Sn alloys.- For a Cu-Sn alloy the observed interference function, I;(K), can be expressed in terms of the partial interference functions by the following equation ;
I;(K) = A I ~ ~ ~ ~ ( K ) + B I ~ ~ ~ ~ ( K ) +
quSn(~)
(2)5. Conclusions.- It has been possible to show that an empirical double hard sphere model can be used to reproduce, within experimental error, features of the observed interference functions of Si, Ge, Ga, Sn, Sb and Bi. Further, it has been shown that the model is consistent with the bond description of the solid state structures. The model may also,be employed to describe alloys of the semimetals Sn with the metals Cu.
R E F E R E N C E S
111
Y. Waseda and K. Suzuki, 2 . Physik B2q, 339 (1975)./2/ J.P. Gabathuler and S. Steeb, 2 . Naturforsch. 34 a, 1314 (1979).
-
/3/ S.P. Isherwood, B.R. Orton and R. Manaila, J. Non-Crystalline Solids,
8-10,
691 (1972). 141 H. Krebs, V.B. Lazarev and L. Winkler, 2 . Anal.Allg. Chem.
353,
277 (1967)./5/ D.I. Page, D.H. Saunderson and C.G. Windsor, J. Phys. C : Solid State Phys.
6,
212 (1973).-
where A = ( c ~ ~ I ~ ~ ) ~ I ( A ~ , B = (CsnfSn)2~(i)2, 161 D.G. Carlson, J. Feder and A. Segmuller, Phys. Rev. A,
2,
400 (1974).c
= 2~;
f / ( i ~ ~ ,F
= c fCu Sn Cu Sn cu cu + 'snfsn' /7/ A. Bizid, L. Bosio, H. Curien, A. Defrain and
M. Dupont, Phys. Stat. Sol.
3,
135 (1974). cCu = atomic fraction of Cu ; fCU = atomic scatte-181
D.M. North, Ph. D Thesis, University of
ring factor of Cu. Sheffield, (1 965).
It has been firmly established by neutron diffrac- /9/ P. Andonov, Rev. Phys. App., 2 , 907 (1974). tion work /34/ that for the Cu-Sn alloy system the /lo/ D. Jovic, J. Phys. C : Solid State Phys.
2,
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455at 3.02
A-1
is equal to the subsidiary peak ~osition (1969)of ISn(K), at 3.02 i-1, this means that IBB(~) could be 1131 Y. Waseda and K. Suzuki, Phys. Stat. Sol.
G,
581 (1971).employed for the two ~artials ICuCu(K) and ICUsn(K>
/I41 W. Knoll and S. Steeb, Phys. and Chem. Liq.,
in the equation above. On this basis a test was
-
4, 39 (1973).made to see if for a Cu 55 at.% Sn alloy 1351 /I51 S.P. Isherwood and B.R. Orton, Phil. Mag.,
15,
could be reproduced by using the same hard sphere 561 (1967).
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'.
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145,
are shown in Figure 3, and in view of the empirical 83 (1966).
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*,
1500 (1975).improvement in the agreement can be obtained if the 1191 B.R. Orton, Z. Naturforsch.
*,
332 (1977).packing density of IBB(K) is reduced to 0.43. /20/ B.R. Orton, Z. Naturforsch. E, 1547 (1979).
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*,
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