Electronic properties of liquid gallium-germanium alloys interpreted with Bachelet, Hamann and Schlutter pseudopotential II

312

Journal of Non-Crystalline Solids 117/118 (1990) 312-315 North-Holland

ELECTRONIC PROPERTIES OF LIQUID GALLIUM-GERMANIUM ALLOYS INTERPRETED WITH BACHELET, HAMANN AND SCHLUTTER PSEUDOPOTENTIAL

II N. KOUBAA, L. ANNO and J.G. GASSER Laboratoire de Physique des Liquides M6talliques Universit6 de Metz. 57045 METZ C6dex 1. FRANCE

ABSTRAI~T In this paper we give our experimental values of the electrical resistivity and thermoelectric power of liquid Ga-Ge alloy. The experimental results are interpreted with first principle model pseudopotentials. We use here two models, the BACHELET, HAMANN and SCHLUTER (B.H.S.) [1] model developed in I [2] and the ANIMALU [3] Simple Model Potential (S.M.P.). I - INTRODUCTIQN The electrical resistivity and the absolute thermoelectric power of liquid gallium, germanium and gallium-germanium alloys have been measured as a function of temperature and composition. No other experimental values have, to our knowledge, been published on this system. The experimental results are interpreted with first principle model pseudopotentials and hard sphere structure factors using the FABERZIMAN [4] formalism. First principle model pseudopotentials are non local and their parameters are energy dependent. In the ahoy one must take into account an energy dependent effective mass, a depletion hole and the Fermi energy core shift. The thermoelectric power includes explicitely an energy dependent contribution The Bachelet Hamann and Schlutter (further called B.H.S. pseudopotential) is energy independent and norm conserving. With theses properties the problem of the depletion hole falls and the pseudopotential is transferable to the alloy case. In our calculations we have used the local screened form factor which we have calculated analytically from B . H . S . pseudopotential [2]. The results are compared to the ANIMALU [3] Simple Model Potential (further called S.M.P.) form factor. 0022-3093/90/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland)

Electrical resistivity According to Z1MAN [5] formula, the electrical resistivity of a pure liquid metal can be written :

p=

3

2 kf

nm2~

o S a(q) v(q) 2 q3 dq 4 e2-t13 k 6 o

[1]

f

We can see that the resitivity depends on two basic quantities, the pseudopotential form factor on the Fermi surface v(q) which is screened by the conduction electrons, and the liquid structure factor a (q). This formula is available for liquid alloys if we replace a(q).v (q)2 (FABERZIMAN [6] ) by: [vl(q)]2 [c(1-c) + c 2 all(q)] + [v2(q)]2 [c(1-c) + (l-c) 2 a22(q)] + 2v 1 (q). v 2 (q). c . (l-c). [a12 (q) - 1]

[2]

where c is the atomic fraction of constituent 1 in the alloy, vi(q) the form factor of specie i in the alloy and the aij (q) are the FABER-ZIMAN partial structure factors [6].

313

N. Koubaa et al./ Liquid gallium-germanium alloys

Thermoelectric power

kf 3 = (3r~2 Z(c)/ff2o(T,c))

The absolute thermoelectric power is proportional to the logarithmic derivative of p with respect to k at kf. In terms of ZIMAN's [5] formula it takes the following form : S=

x2k 2 T~, b 3 eEl

[7]

where Z(c) and ~ o ( T , c ) are respectively the mean valence and the mean atomic volume of the alloy obtained by a linear interpolation of the pure metal values.

[3] III - R E S U L T S AND DISCUSSIONS

Here Z is the thermoelectric parameter which is given by : kf

[41 k=kr

with: Z = 3 - 2 q - d2 where 3 is the free electron contribution, 2q the pseudopotential term and r/2 the energy dependent term which is equal to zero for energy independent model potentials. Our calculations have been performed with ASHCROFF-LANGRETH [7] hard sphere partial structure factors for alloys. We choose, as parameters, two hard sphere diameters for the two species and held them constant with concentration but not with temperature. The hard sphere diameters have been obtained from the pure metal data. At each temperature, the hard sphere diameters are deduced from the experimental densities of the pure metals, compiled by CRAWLEY [8] and from the packing fraction given by the W A S E D A empirical law [9]. Hi (T) = A i exp (-BIT)

The techniques of measurements of the resistivity and of the thermopower have been described by ANNO [10] and MAYOUFI [11]. RESISTIVITY: On figure 1 we have reported the resistivities of the Ga-Ge system as a function of temperature. As can be observed the resistivity is a linear function of the temperature for all the Ga-Ge alloys. We have represented on figure 2 the resistivity of the Ga-Ge system as a function of the Ge concentration at 1000°C. The calculated resistivities (curve 1 and 2) are slightly different from the experimental ones (curve 3). 80 Ge

60 :~ ~

G Ga70Ge30C'~J/

40

[5] T(°C)

where the parameters A i and B i have been taken from WASEDA's book [9]. The hard sphere diameters are : i 3 (T) = [ 6 rli(T) ~2o(T)/r~]

[61

In the alloy we have taken into account the modification of the mean atomic volume f~o(T) and of the Fermi wavevector kf

20

' ' 0 400 800 1200 Figure 1 Resistivity of liquid Ga-Ge alloy

In the calculation we have used the B.H.S. form factor which we have determined analytically [2]. The two models were screened linearily by the Vashishta-Singwi [12] dielectric function. We have also reported on figure 2 the resistivity of pure g e r m a n i u m computed with the

N. Koubaa et M./Liquid gMh'um-germanium alloys

314

pseudopotentials and the experimental structure factors (triangle: B.H.S.; cross: S.M.P.).

70

I

I

I

Figure 3 shows that the Ga-Ge experimental thermoelectric power has a comparable temperature dependency for all the concentrations. On figure 4 we can observe that the experimental thermoelectric power does not vary significantly with concentration

I

[]

THERMOELECTRIC POWER

[]

p3) []

(2) E

o ;}

:=k

o~

@ g~

5O (1) BHS

-1 "

.m 1,1

(2) SMP (3) Exp

@

n

0,0

I

0,2

n

I

0,4

n

I

0,6

-2

E

(at.%Ge) 40

n

0,8

1,0

g~ [-

Figure 2 Resistivity of liquid Ga-Ge alloy as a function of Ge concentration -3

The resitivities computed with B.H.S. model pseudopotential are close to those computed with S.M.P. model, but the two models underestimate the resistivity of liquid germanium and overestimate the resistivity of liquid Gallium. H o w e v e r the results are sensitively different with the experimental structure factors of Ga and Ge which present a shoulder on the high angle side of their first peak. We can observe that, with the experimental structure factors, we improve the resistivity computed with the two model-potentials for both Ge and Ga. We have computed the resistivity of the Ga-Ge alloy at 1000°C. Our results with B . H . S . [2] and S.M.P. [3] form factors are compared to the experimental resistivity. Fig. 2 shows that our calculated values are slightly different from the experimental ones. In the case of alloys, the partial experimental structure factors are not known. However, with hard spheres the agreement with experimental data is satisfactory but could probably be improved with experimental structure factors.

i

Gag0Gel0 GaSIX3e20 Ga70Ge30 Ga60Ge40 Ga50Ge50 Ga40Ge60 Ga30Ge70 • Ga20Ge80 4- Gal0Ge90 • Ge . . . .

!

. . . .

A~l • T(°C) i

250

. . . .

i

500

. . . .

|

!

750

1000 Figure 3 Thermoelectric power of liquid Ga-Ge liquid alloys

I

?

I

Ga-Ge alloy

I

I

T=800°C

t-

@

@ \

12l

(]) BHS (2) SMP

[-

(3) Exp (at.%Ge) -5

i

i

i

i

0,0 0,2 0,4 0,6 0,8 1,0 Figure 4 Thermoelectric power of liquid Ga-Ge alloy versus Ge concentration

On figure 4 we have plotted the thermoelectric power of the Ga-Ge alloy at 800°C. We compare

N. Koubaa et al./ Liquid gallium-germanium alloys

the experimental results to those computed with the pseudopotential form factors (B.H.S. and S.M.P.). For pure gallium, both form factors give results very near the experimental thermopower (at less than 1.5 ~tV K -1 ). For pure germanium, the B.H.S. thermopower is at less than 0.5 lxVK -1 from the experimental value while the S.M.P. is at about 3.0 ~tVK -1. The influence of the experimental structure factor on the thermopower of the pure metals is very small ( less than 0.2 p.V K -1 ). For the alloy, the results obtained with B.H.S. form factor are near the experimental thermopower (at less than 2.0 p.V K -1 ). The results are not so good with the S.M.P. form factor which gives results at about 4.5 IxV K -1 . It is probably due to the neglected energy dependent contribution which does not appear with B . H . S . form factor. However, the divergence between the three curves is not very important since the scale is very expanded. CONCLUSIONS B.H.S. and S.M.P. local screened model (pseudo)potentiel were used for computing the resistivity and the thermoelectric power of gallium, germanium and of the Ga-Ge system. The resistivity is well explained with these model-potentials. It is probable that the remaining discrepancy comes from the local screening and from the hard sphere structure factors. Indeed, for pure Germanium and Gallium, the experimental structure factor improves the resistivity Concerning the thermoelectric power, the non local contribution can be important and the use of first principle energy independent form

315

factors represents a progress in liquid alloys electronic properties calculations. Our results are very satisfactory and confirm that the B.I-I.S. form factor is probably more adequate for describing the resistivity and the thermoelectric power of liquid Ga-Ge alloy.

1) BACHELET G.B., HAMANN D.R. and S C H L U T E R M.; Phys. Rev. B 2 6 (1982) 4199. 2) KOUBAA N. and GASSER J.G. ; Analytical screened form factor obtained from Bachelet, Hamann and Schluter pseudopotential. This volume 3) ANIMALU A.O.E. ; Phil. Mag. 11 (1965) 379. 4) FABER T.E. and ZIMAN J.M. ; Phil. Mag. 11 (1965) 163. 5) ZIMAN J.M. ; Phil. Mag. 6 (1961) 1013. 6) FABER T.E. ; Introduction to the theory of liquid metals ; (1972) Cambridge at the University press. 7) ASHCROFT N. W. and LANGRETH D. E.; Phys. Rev. 156 (1967) 685. 8) CRAWLEY A.F.; Int. Met. Rev. 1___99(1974) 32. 9) W A S E D A Y. ; The structure of non-crystalline materials (1980) Mac Graw Hill. int. book company p 59. 10) ANNO L. ; M6moire d'ingrnieur CNAM (1985). Universit6 de Metz (France). 11) MAYOUFI M. ; th~se (1985). Universit6 de Metz (France). 12) VASHISHTA P. and SINGWI K.S. ; Phys. Rev. B___6_6(1972) 875.