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.
<10.1051/jphyscol:1980871>. <jpa-00220524>
HAL Id: jpa-00220524
https://hal.archives-ouvertes.fr/jpa-00220524
Submitted on 1 Jan 1980
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
CoZzoqtre C 8 , suppZ&ment au n08, Tome 4 1 , aoiit 1980, page C8-280
JOURNAL DE PHYSIQUE
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 separation. (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
The values of aA, oB, qA, '1B and cA were vat-ied
diffraction measurements on the moltensemiconduc-
in a systematic way until the best possible agree-
tors, Si /1,2/ and Ge /1,2,3,4/ and the semimetals
ment was achieved. The values of these parameters
Ga /5,6,7/,
are recorded in table 1.
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
I
(K) and I (K) are shown in Figs. 1 and 2 particuexP
T
lar note should be taken of the excellent agreement
of incident radiation) which have main peaks which
are broadened by a subsidiary maximum or shoulder
on the high K side.
Agreement for semiconductors was not so good. It was
found possible to match the shape of the peak or
The model proposed for liquid semiconductors 1181
and semimetals /19/, 1201, /21/ contains the £0110wing features. (a) The atoms of these elements can
obtained over the whole of the main peak of IGa(K).
height, for Si, but not both together. It may also
be noted that the general agreement over the second
and subsequent peaks was poor.
come together to give either long A or short B distances 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
3. Discussion of the results.- From the evidence of
1181 that it is possible to divide the liquid up
the results considered above it is clear the the
into partial interference functions IAA(K)
molten elements Si joins, Ge, Ga, Sn, Sb and Bi as
and
IBB(K) and express the total interference function
describable in terms of a double hard sphere model.
as the superposition of only two partial functions ;
The nature of the atomic interactions which brings
IT (K) = cA ( 1+cB)IAA(K)
2
+ cBIBB(K)
(1
this about will now be discussed.
Table 1 gives the crystal structures of the elements
cA = atomic fraction of A component
2. Results.- The experimental data which were used
in this work were chosen from the author's own measurements 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)
I(K)
to the experimental
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
the packing density.
hard sphere diameter and
mentioned in this work. They either show strong
covalent bonds or some evidence of p type directional 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'directional 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 interatomic 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
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980871
surrounding atoms do not take up the short distance
but stay at the longer distance because it is unlikely that the orientation of the p type orbitals
will be correct. The longer A separations would
correspond to non-directional metallic bonds. Measurements 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
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.
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 potentials 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)
=
(2)
A I ~ ~ ~ ~ +( KB )I ~ ~ ~ ~ +( K
quSn(~)
)
where A = ( c ~ ~ I ~ ~ ) ~ I B( A= ~(CsnfSn)2~(i)2,
,
c
=
cCu
;
2~Cu Sn CufSn/ ( i ~ ~ F, = ccufcu
=
+
'snfsn'
atomic fraction of Cu ; fCU = atomic scatte-
ring factor of Cu.
It has been firmly established by neutron diffrac-
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).
161 D.G. Carlson, J. Feder and A. Segmuller,
Phys. Rev. A, 2,400 (1974).
-
/7/ A. Bizid, L. Bosio, H. Curien, A. Defrain and
M. Dupont, Phys. Stat. Sol. 3,
135 (1974).
181 D.M. North, Ph. D Thesis, University of
Sheffield, (1 965).
/9/ P. Andonov, Rev. Phys. App., 2, 907 (1974).
/lo/ D. Jovic, J. Phys. C : Solid State Phys. 2,
I135 (1976).
partial interference functions do not depend on
/]I/
H.U.
Gruber and H. Krebs, Zeit. Anog. Allgemeine
concentration and moreover, ICuSn(K) is similar to
Chem., 369, 194 ( 1969).
lCu~u(K)' Since the first peak position of ICUCU(K)y
1121 H. Krebs, J. Non-Crystalline Solids, 1,455
at 3.02 A-1 is equal to the subsidiary peak ~osition
(1969)
tion work /34/ that for the Cu-Sn alloy system the
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 ~artialsICuCu(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).
-
of ISn(K),
made to see if
for a Cu 55 at.% Sn alloy 1351
could be reproduced by using the same hard sphere
interference functions, in equatipns 2, as was utilized for the successful model for Sn. The results
/I51 S.P. Isherwood and B.R. Orton, Phil. Mag.,
561 (1967).
'I6/ P. Lamparter* S. Steeb and W.
Naturforsch. ,
&
3
90 (1976).
15,
'.
145,
are shown in Figure 3, and in view of the empirical
1171 N.W. Ashcroft and J. Lekner, Phys. Rev.
83 (1966).
nature of the model, the agreement is good. An
improvement in the agreement can be obtained if the
I181 B.R. Orton, 2. Naturforsch.
1191 B.R. Orton, Z. Naturforsch.
packing density of IBB(K) is reduced to 0.43.
/20/ B.R. Orton, Z. Naturforsch.
On the basis of the model proposed for pure metals,
/21/ B.R. Orton,
this result could be interpreted as evidence for Cu
and Sn coming together with a short directional p
/22/ C.N.J. Wagner and N.C. Halder, Adv. Phys., 16,
241 (1967).
1231 H. Krebs, Fundamentals of Inorganic Crystal
Chemistry, McGraw Hill, London 1968.
type bonding.
2.
Naturforsch.
*,
*,
E,
*,
1500 (1975).
332 (1977).
1547 (1979).
397 (1976).
JOURNAL DE PHYSIQUE
C8-282
1241 R.W. Wyckoff, C r y s t a l S t r u c t u r e s , 2nd Ed.,
I n t e r s c i e n c e , New York, 1965 p. 22.
1251 C. N o r r i s and J.T.M. Wotherspoon, J . Phys. F :
Metal Phys. 1, 1599, (1977).
1261 F. G r e u t e r and P. Oelhafen, 2 . Physik B,
123 (1979).
T A B L E
Element Hard s p h e r e parameters
24,
/27/ J.S.P. C a s t e l i j n s , H.W.J.M.
a a n de Brugh and
A.R. Vroomen, J. Phys. F : Metal Phys. 1,
2457 (1977).
I
uA OB
51
"
rjB
cA
'crystal structufe
Type
dist.
N.N.
S
Si*
2.42 1.88 0.42 0.42 0.44 A4
0.38 0.38 .40 Cubic
2.35
1281 C. N o r r i s , D.C. Rodway and G.P. Williams, 2nd
I n t . Conference o n t h e P r o p e r t i e s of Liquid
Metals, Ed. S. Takeuchi, 1972 p . 181.
Ge**
2.70 2.07 0 . 4 1 0.41 0.42 A4
Cubic
2.44
1291 Y . Baer and H.P. Myers, S o l i d S t a t e Comunicat i o n s 2, 833 ( 1977).
Ga
2.78 2.25 0.50 0.47 0.49 Orthorhombic,
2.44 and s i x atoms
i n p a i r s between
2.71 and 2.80
Sn
3.03 2.42 0.48 0.47 0.67 A5
Tetragonal, four
a t 3.105 two a t
3.175
Sb
3.11 2.33 0 . 4 1 0.41 0.60 A7
Hexagonal, t h r e e
a t 2.90, t h r e e
a t 3.36.
Bi
3.23 2.45 0.43 0.40 0.60 A7
Hexagonal, 3.105
3.474.
.
1301 M. S i l b e r t and W.H. Young, Phys. L e t t e r s , 5&,
469 (1976).
/31/ D. Levesque and J.J. Weis, Phys. L e t t e r s ,
473 (1977).
1321 R. Oberle and H. Beck, S o l i d S t a t e Comunicat i o n s , 32, 959 (1 979).
1331 K.K. Mon, N.W. A s h c r o f t and G.V.
Rev. B, 19, 5103 (1979).
C h e s t e r , Phys.
1341 J.E. Enderby, D.M. North and P.A. E g e l s t a f f ,
Adv. Phys., 16, 171 (1967).
1351 D.N. North and C.N.J. Wagner, Phys. and Chem.
of Liq., 2, 87 (1970).
*
Higher packing d e n s i t i e s gave peak h e i g h t
c o r r e c t , lower gave c o r r e c t shape.
**
cA i n c o r r e c t l y quoted a s 0.58 i n
[18]
.
FIGURE 3. i(K) Cu 55 at.% Sn.
EX^^) I, (K)
- - - -;]cu(K),7 ~ 0 . 4 7 5
5 (K),?n=0475,7a=0,470.
I, (K),-.-;I~~(K),~~ 0 , 4 3 0
Is,(K), 7*=0475,7, = 0.430.