1239 obtained with hard sphere structure factors, curve 1 and 2 with the t matrix form factor computed respectively with Waseda's 8 and Ballentine et a125 phase.
Journal of Non-Crystalline Solids 61 & 62 (1984) 1237-1242 North-Holland, Amsterdam
1237
ELECTRICAL RESISTIVITY OF BISrIUTH AND BISrIUTH-GER~IANIUM ALLOYS IN THE LIQUID STATE J.G. GASSER, rl. MAYOUFI, G. GINTER AND R. KLEIM. Centre "~lati~re-Rayonnement-Structure", Facult~ des Sciences, l l e du Saulcy, 57045 METZ C~dex, FRANCE. The e l e c t r i c a l r e s i s t i v i t y of I I bismuth-germanium alloys has been measured from the melting point to I150°C. The temperature dependence of the pure components are discussed in the framework of Ziman formalism with experimental interference functions and d i f f e r e n t model pseudopotentials (HeineAbarenkov, Animalu-Heine, Shaw-Hallers) and with the "extended Ziman Formula" by using the t matrix expressed in term of phase s h i f t s . The composition dependence is i n t e r p r e t e d with the same model pseudopotent i a l s and phase s h i f t s . But no experimental p a r t i a l structure factors being a v a i l a b l e , we used Ashcroft and Langreth hard sphere p a r t i a l structure factors. The experimental values are well reproduced f o r the a l l o y with the form factors giving good results f o r the pure metals. I . ItlTRODUCTION I t has been l a r g e l y admitted that the problem concerning simple l i q u i d metals has been well resolved from a t h e o r e t i c a l point of view.
In these l i q u i d a l l o y s ,
the transport properties are well described by the nearly free electron theory with Ziman's formula 1'2 in the case of pure metals and with Faber-Ziman's 3 one (1965) for a l l o y s ,
However the s i t u a t i o n is le'ss clear for semi-metals and
t h e i r alloys from a q u a n t i t a t i v e point of view.
The e l e c t r i c a l r e s i s t i v i t y of
BixGe(l_x ) alloys has to our knowledge never been determined experimentally. However the r e s i s t i v i t y
and thermopower of alloys of metals of valency 4 and 5
have been studied in our laboratory 4'5.
For this kind of alloys i t is i n t e r -
esting to search i f there exists a negative temperature c o e f f i c i e n t of the resistivity
l i k e in Ge-Sb6 and the Ziman formula can be employed though the
mean free path is small. In pure l i q u i d metals, the experimental structure f a c t o r s , measured e i t h e r by X-ray or by neutron s c a t t e r i n g , are in general a v a i l a b l e .
They can also
be computed, assuming the l i q u i d to be an assembly of hard spheres 7.
The
packing f r a c t i o n ~ can be taken from Waseda's empirical law8. = A exp(-BT)
(I)
where A and B are tabulated in Haseda's book8. The p a r t i a l interference functions are scarcely available from the experiment.
We used again hard spheres 9,
At each temperature the hard sphere
diameters are deduced from the packing fractions and the experimental densities 0022-3093/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Z G. Gasser et al. / Electrical resistivity o f bismuth and bismuth-germanium alloys
1238
of the pure components obtained respectively from equation ( I ) and from the densities compiled by Crawley I0.
They are held constant in the a l l o y .
Different form factors have been used in this work: - l o c a l one's l i k e the Harrison Point lon Potential ( P . I . p . ) I I ;
the Ashcroft
Empty Core Potential (E.C.P.)]2; the Shaw local Optimised Model Potential (O.M.P.) 13.
Their parameters have been f i t t e d on the node of the non local
Animalu-Heine Model Potential (A.H.M.P.114-17 according
to a procedure
described by Gasser 6. -the non local but local screened Heine-Abarenkov - Animalu Simple Model Potential (S.M.P.) 15'17, -
f u l l non local model potentials l i k e the Animalu-Heine Model Potential 16
the Shaw Model Potential 13 and the Shaw-Hallers Model P o t e n t i a l . Analytical expressions e x i s t f o r the O.M.P., the E.C.P., the P.I.P. and the S.M.P..
The l a s t mode] has the advantage to have parameters f i t t e d d i r e c t l y on
the spectroscopic data of ions.
We used A.H.M.P. under i t s tabulated form in
Harrison's book I I , and renormalized the form f a c t o r on the atomic volume of the metal at the considered temperature f o l l o w i n g the procedure discussed by Ziman18 The Shaw non local form f a c t o r is e n t i r e l y computed following Shaw's o r i g i n a l paper 13 corrected from the e f f e c t i v e masses19 with a uniform depletion density d i s t r i b u t i o n . The parameters are computed at the l i q u i d density s t a r t i n g from Ese and Reissland 20 data. A difficult
problem is to t r e a t c o r r e c t l y the energy dependence of the para-
meters in t h e ~ l o y .
At our knowledge the Shaw-Hallers (we are g r a t e f u l to
Dr. Van der Marel for making available Shaw-Hallers computer program) potential is the only one in which this dependencein taken into account. We have also computed the Shaw-Hailers form factor at the corresponding liquid densities. In all cases (except the A.H.M.P. tabulated form factor) we used the Vashishta-Singwi dielectric screening function21. An alternative solution to model potential treatment has been proposed by Evans et a122 for pure metals and by Dreirach et a123 for alloys. The model potential form factor is simply replaced by a t matrix form factor in Ziman's formula. I t can be expressed in terms of phase shifts who are computed from a muffin tin potential constructed either by the method of Dreirach et a123 or by that described by Mukhopadhyayaet a124. For semi-metals i t seems that this second approach gives in general better results 5. 2. ELECTRICALRESISTIVITYOF PUREBISMUTII ~le have reported in Fig. 1 the experimental electrical r e s i s t i v i t y of Bismuth ( f u l l points) together with different calculated curves. Full curves are
J. G. Gasser et al. / Electrical resistivity o f bismuth and bismuth-germanium alloys
1239
obtained with hard sphere s t r u c t u r e f a c t o r s , curve 1 and 2 with the t m a t r i x form f a c t o r computed r e s p e c t i v e l y w i t h Waseda's 8 and B a l l e n t i n e et a125 phase shifts.
Curve 3, 4, 5 and 6 are obtained r e s p e c t i v e l y w i t h the P . I . P . , the
E.C.P., the S.M.P. and the O.M.P. form f a c t o r s .
EISMUTH
.* t~;O
•
,;"
• t*
140
"~ 120
I) Waseda's phase s h i f t s 2) B a l l e n t i n e ' s phase s h i f t s 3) Shaw-Hallers potential 4) A,-H.M.P. 5) Shaw optm. potential
too 9o ~o 7Q s( 4o
/
/, ~ 2
~1oo
I0~~
~o
~
0
EO
lO
~."
20
~
4,O
q(A -I )
Tfc)
Figure 1
Figure 2
R e s i s t i v i t y of l i q u i d bismuth points are the experimental values
Integrands of the r e s i s t i v i t y d i f f e r e n t form f a c t o r s
with
I n d i v i d u a l points are c a l c u l a t e d with the experimental s t r u c t u r e f a c t o r s t a b u l a t e d in Waseda's book ( " r e c t a n g l e s " with Waseda's phase s h i f t s ,
"tri-
angles" w i t h B a l l e n t i n e ' s one, "open c i r c l e s " with the S.M.P., "plus with E.C.P.) With the experimental s t r u c t u r e f a c t o r s , the r e s i s t i v i t y coefficient
are in general smaller than w i t h hard spheres.
in t a b l e 1 the r e s i s t i v i t y
and i t s temperature We have reported
obtained at 300 ° with Waseda's experimental s t r u c -
ture f a c t o r Ithe experimental r e s i s t i v i t y
is 129.0 u~.cm).
TABLE I : ! .
! Phase S h i f t s i B a l l . v Was. . . .
i_ _
! i
L ! i OMP t . .
PIP .
L i SMP .
L ! !Shaw- ! i AHMP i Shaw I E Haller~ L
L--L
53,2 ! 101,0 !
81,2 ! 81,1
L 57,7
L 59,1 !
122,9 ! 122,9 L 84,3 L 46,2 ! 110,2 !
88,9 [ 88,7
! 64,1
! 64,3 !
L - -
! - -
!
!a(qlexp!" " 119,5 ! 124,8 ! !a(q)H.S!
.
ECP
. L - -
76,0 !
!
!.--!
1240
J.G. Gasser et al. / Electrical resistivity o f bismuth and bismu th-germanium alloys
The t matrix formulation is in r e l a t i v e l y good accordance with the experimental value near the melting point while "model pseudopotentials" give in general smaller values.
We have examined in detail the integrand of the + are reported on f i g . 2.
r e s i s t i v i t y of bismuth at 30O°C. Different curves
The r e s i s t i v i t y is proportional to the area under these curves.
I t appears
clearly that the different form factors give very different curves even i f the r e s i s t i v i t y is the same (Shaw and Shaw-Hailers form factors look very different but the r e s i s t i v i t y is at .6 % the same). Two regions can be observed, below the node in the region of the f i r s t peak of the structure factor and above the node, near 2 kf.
The t matrix gives in general a larger contribution of
the second region than model potentials.
For example BallentineJs phase shifts
give a very similar curve to that obtained with the S.M.P. below the node. The supplementary contribution of the t matrix form factor occurs above the node. Non-local screening is not very important for high valency metals as has been emphasized by Animalu14.
The A.H.M.P. and S.M.P. curves are indeed
confounded. However,the use of the Vashishta-Singwi d i e l e c t r i c screening function would enhance the A.H.M.P. curve of about lO % (according to our calculations).
The r e s i s t i v i t y , though very sensitive to the details of the
form factor is an integral value.
I t is necessary to test also model poten-
t i a l s on other physical properties of the liquid metals l i k e the temperature coefficient of the r e s i s t i v i t y or the thermopower. 3. ELECTRICAL RESISTIVITY OF GERMANIUM-BIS~IUTH ALLOYS We have reported on f i g . 3 our experimental measurementas a function of temperature and on f i g . 4 as a function of concentration at lO00°C together with the calculated curves.
Curve l is obtained with the S.M.P., curve 2, 3,
4, 5 with the phase shifts respectively of Waseda8 for Ge and Bi, of Dreirach et a123 for Ge and Ballentine et a125 for Bi, of Dreirach et al for Ge and Waseda for Bi, of Waseda for Ge and BaIlentine et al for Bi.
Curve 6 is ob-
tained with the Shaw-Hailers non-local model-potential and takes into account the energy dependence of the parameters in the a l l o y . the Shaw non-local optimised model potential.
Curve 7 is obtained with
The parameters are determined
for the pure liquids at lO00°C. For the a l l o y the form factors are corrected for volume and kf variations. As can be observed on f i g . 4, the best results are obtained with phase shifts Model potentials though very sophisticated cannot reproduce correctly the experi mental curve.
J. G. Gasser et al. / Electrical resistivity o f bismuth and bismuth-germanium alloys
RESISTIVITY OF Bi Gr ALLOYSAS A
1GS
.Bi,Ge. ,~ •
-"
-•
,Q.~
"" ""i"
- . . . . • ° ' °o ". , , ~ l . o e
•
...,,,,,.,,
•
•
~
x,,"~m9
.o . , ~ "
.,,~ ,,
o,,,
145
1241
• •
• ""
,ooo~ •
125
. •
o,
,,
•
• , . oo
• •
•
.e
.E"
e~
.5"
• •
,4.'
,3' Q,o q. • ~11,*
,I,,
"
"'"
• 02
,,''
° " ' ° "
'~lql'~
~S
el woo
8~
oB * ' ' °
°i"
" *°"
° " Gr
9 0
IOQO
1180
T('C)
lO 20
30
40
SO
GO
70
EO SO
me'/, Bi
FIGURE 3
FIGURE 4
Temperature dependence of the e l e c t r i c a l r e s i s t i v i t y of l i q u i d Bi-Ge a l l o y s .
Concentration dependence of the e l e c t r i c a l r e s i s t i v i t y of Bi-Ge alloys.
4. CONCLUSION In some sense, the fact that the t matrix formulation does not need the Born approximation could perhaps explain why i t gives b e t t e r results traduced on the form f a c t o r near 2 kf.
However, the bismuth mean free path is only 1.7 times
the interatomic distance.
I t is probable that m u l t i p l e scattering would also
occur and that we are at the l i m i t of v a l i d i t y of the d i f f r a c t i o n model. REFERENCES I) J.tl. Ziman, Phil. Mag. 6 (1961) 1013. 2) C.C. Bradley, T.E. Faber, E.G. Wilson and J.PI. Ziman, Phil. Mag. 7 (1962) 865. 3) T.E. Faber and J.M. Ziman, Phil. Mag. I I (1965} 153. 4) A. Bath, J.G. Gasser, J.L. Bretonnet, R. Bianchin and R. Kleim, Phys. Colloque C41 I1980) 519. 5) J.G. Gasser, Th~se de Doctorat d ' E t a t : "Contribution ~ l'~tude des propri~t~s ~lectroniques, r ~ s i s t i v i t ~ et pouvoir thermoelectrique dWalliages de m~taux polyvalents et de t r a n s i t i o n b l ' ~ t a t l i q u i d e . (Ge-Sb, Pb-Sb, Mn-Sb, Mn-Sn, Mn-ln et Mn-Zn) 17 D~cembre 1982. (Universit~ de METZ). 6) J.G. Gasser and J.D. Muller, Nato summer school on l i q u i d and amorphous metals, Edited by E. L~scher and H. Coufal, S i j t h o f f & Noordhoff (1980) 631. 7) N.W. Ashcroft and J. Lekner, Phys. Rev. 14b (1966} 83.
1242
J.G. Gasser et aL / Electrical resistivity o f bismuth and bismuth-germanium alloys
8) Y. Waseda, The structure of non-crystalline materials (1980), edited by McGraw Hill int. book company 9) N.W. Ashcroft and D.C. Langreth, Phys. Rev. 156 ((967) 685. I0) A.F. Crawley, Int. Met. Rev. 19 (1974) 32. I I ) W.A° Harrison, (1966) Pseudopotentials in the theory of metals, edited by Benjamin Inc. New York. 12) N.W. Ashcroft, Phys° Lett. 23 (1966) 48. 13) R.W. Shaw, Phys. Rev. 174 (1968) 769. 14) A.O.E. Animalu, Phil Mag. II (1965) 529. 15) A.O.E. Animalu, Phys. Rev. B8 (1973) 3542. 16) A.O.E. Animalu and V. Heine, Phil. Mag. 12 (1965) 529. 17) V. Heine and I.V. Abarenkov, Phil. Mag. 9 (1964) 451. 18) J.M. Ziman, Adv. Phys. 16 (1967) 551. 19) 20) 21) 22) 23)
R.W. Shaw, J. Phys. C2 (1969) 2350, O. Ese and J.A. Reissland, J. Phys. F3 (1973) 2066. P. Vashishta and K.S. Singwi, Phys. Rev. B6 (1972) 875. R. Evans, D.A. Greenwood and P. Lloyd, Phys. Lett. 35A (1971) 57. O. Dreirach, R. Evans H.J. GUntherodt and H.U. Kunzi, J. Phys. F2 (1972) 709. 24) G. Mukhopadhyaya, A. Jain and V.K. Ratti, Solid State com. 13 (1973) 1623, 25) L.E. Ballentine and M. Huberman, J. Phys. ClO (1977} 4991. +
Computer plotted curves available on request.
