Short Notes K131

phys. stat. sol. (b) 139, K131(1987) Subject classification: "2.15; S4

Department of Physics, Sabour College (a) and Marwari College, Bhagalpur University1) (b)

Electronic Transport Properties of Gallium: Harrison's Method

BY M. MEHAR KHAN (a) and S.M. RAFIQUE (b)

Gallium is an interesting metal both from theoretical and experimental points of view because it shows an anomalous structural behaviour. Although its elec- tr ical resistivity is only 25.8 u n c m , it has got its place in the semimetal group, Unlike other simple metals it shrinks on melting and has slightly more ordered structure just above its melting point, There a r e subtle structural distinctions in comparison to other liquid metals which is also exhibited by a shoulder on the right side of the main peak of the interference function (Asca- relli /I/). Though the system exhibits many interesting physical properties , the theoretical study of the electronic transport in Ga is very scarce. We, therefore, propose to study it through Harrison's f i r s t principle pseudopoten- tial technique.

The computation of the electronic transport properties needs a precise knowledge of the pseudopotential matrix element ( $+ 3 Iw I k ) commonly known as the form factor. It may be mentioned that the other important in- gredient, i. e, the structure factor a(:) is experimentally available from X-ray o r neutron diffraction measurements. The Harrison method /2/ based on a rigorous mathematical approach seems to be more reliable in providing the true crystal potential. Herein, the non-local screened form factor is represented

by

+

(1) -* -b R < k + d I w ~ k ) = v ~ + V c * + V d * + v ~ + W ,

where the significance of different terms and their computational details have been discussed elsewhere /3,4/. The form factor is in turn used to compute the electrical resistivity and thermopower through the well known Ziman and Bradley formulae, respectively, on the lines of Mitra et al. /5/. The structure

factor needed here has been taken from Page et al. /6/.

1) Bhagalpur City, Bihar 812002, India.

K132 physica status solidi @) 139

Fig. 1. Plot 9f form factor (in J) versus '1 = q/kF for different values of t heXa - exchange parameter. The data available from fittings to the Fermi surface and other properties taken from /8/ have been

--- 2/3 indicated by 0 and A. a = -as,--- 1,

In previous communications /3,4/ we

I

have shown that the eigenvalues have a priority in these computations and there i

- 0 7 ; ' I ' ' I ' " 04 I t l6 2n exists a controversy in their choice.How-

ever, if a proper choice is made no arbi- II-

t rary parametrization is necessary to predict acceptable theoretical results. With this view, Harrison's method /2/ has been applied to liquid gallium

near its metling point so as to observe how far our previous interferences re- garding other metals are justified to this peculiar metal. For the purpose dif- ferent sets of available eigenvalues have been tried and the best result obtained through the eigenvalues of Clementi /7/ is being reported. These eigenvalues provide a form factor (Fig. 1) which agrees with sufficient experimental data /8/ and predicts satisfactory results of electrical resistivity and thermopower.

To show the superiority of the Slater value of the XOL exchange parameter

over other values i.e. 4c = 1 and a= 213 suggested by different authors /9 to 12/ we have repeated our computations with these parameters also (see Table 1

and Fig. 1). The impact of popular forms of exchange correlation may be found in Table 1 and 2 summarising the result of Hubbard-Sham (H-S), Kleinman- Langreth (K-L), Vashishta-Singwi (V-S), and Shaw forms of the exchange cor- relations. In Table 1 and Fig. 1 the experimental data have been supplied for comparison. The data of form factors taken from /8/ have been obtained through fittings to different physical properties of the metal.

values and the (V-S) form of dielectric screening differ markedly for of = 1 and or. = 0.70644 (Slater parameter). The latter differs inappreciably from that with a = 2/3 due to small variation in a. Even then the deviations in the elec- t r ical resistivity are significant (eL = 32.OpQcm with a= 2/3, QL =25.3~&m

with oc = 0.70644, and qL = 47.6pQcm with oc = 1). In comparison to the ex-

A perusal of Fig. 1 reveals that the form factors using Clementi's eigen-

Short Notes

T a b l e 1

(H-S)

22.96

- 0.06

K133

(K-L) Shaw exper.

23-46 29.42 25.8

- 0.08 - 0.13 - 0.4

Computed electrical resistivity QL (inpSlcm) and thermoelectric power &L (in pV/K) using the present form factors

25.36 QL (P Qcm)

QL - 0.105 (PV /K)

lexehanged correlatio W-s)

47.57 32.02

- -

perimental value of 25.8 pch cm the Slater parameter yields the best result, Its

superiority is further substantiated by the fact it gives a form factor which agrees with a number of points available from fitting to different experimental observations such as Fermi surface, optical, and other properties of the metal. However, all these form factors reproduce the correct sign of the thermopower and give a qualitative agreement (vide Table 1).

From the inspection of Table 1 and 2 we observe that the different forms of exchange-correlation function produce smaller variations in form factor spe- cially in the region 1 5 q 2 2 which is the most contributing region because the peak of the structure factor lies in it. Consequently, the electrical resistivities and thermopowers thus computed lie in the vicinity of each other, the largest departure being with the Shaw form of the dielectric screening, which worsens the electrical resistivity and improves the thermoelectric power but the wor- sening is more prominent.

The present analysis shows that like other simple metals studied earlier, the peculiar metal Ga also follows the same pattern as far as the electronic transport is concerned. The sensitivity of the form factor and the computed

properties to the choice of eigenvalues, the superiority of the Slater parameter,

K134

T a b l e 2

p hysica status solidi (b) 139

(K-L)

- 0.5207

- 0.6052 - 0.5415 - 0.5112 - 0.3941 - 0.3137 - 0.1922

- 0.1178 - 0.0367 + 0.0076

+ 0.0508

Form factors with (V-S) , (H-S) , (K-L), and Shaw forms of exchange correlation

Shaw

- 0,5207

- 0.6124 - 0.5670 - 0.5560

- 0.4496 - 0.3600 - 0.2258 - 0.1288 - 0.0401

+ 0.0139

+ 0.0511

3.0

D.2 D.4 D. 6

D. 8 1.0 1.2

1.4

1.6 1.8

2.0

form factors ( 2 . 4 2 5 ~ l O - l ~ J)

W-S)

- 0.5207

- 0.6073 - 0.5489 - 0.5244 - 0.4409 - 0.3082 - 0.2034 - 0.1217 - 0.0380 + 0.0101

+ 0.0509

:H - S) - 0,5207

- 0.6064 - 0.5450 - 0.5154 - 0.3966 - 0.3133 - 0.1899 - 0.1162

- 0.0359 - 0,0051

+ 0.0507

and the second-order effect of the exchange-correlation functions are equally valid in this case, too.

One of the authors (S.M.R.) is thankful to the U.G.C., New Delhi for f i -

nancial assistance. The authors acknowledge with reverence the guidance and encouragements from Dr. P. L. Srivastava, Dr. N.R. Mitra, Dr. K.K. Muk- herjee, Dr. BOB. Sahay, and the valuable cooperation of Dr. R.N. Singh. We a r e thankful to Dr. Bechan, Principal, Marwari College, Bhagalpur for his

generous support.

References /1/ P. ASCARELLI, Phys. Rev. - 143, 36 (1966). /2/ W.A. HARRISON, Pseudopotential in the Theory of Metals, Benjamin, Inc,,

/3/ S.M. RAFIQUE, N.R. MITRA, and P.L. SRIVASTAVA, phys. stat. sol. @I)

New York 1966.

119, - K113 (1983).

/4/ B.N. SINGH and S.M. RAFIQUE, phys. stat. sol. @), to be published.

Short Notes K135

/5/ N.R. MITRA, J. THAKUR, and P.L. SRIVASTAVA, phys. stat. sol. (b)

/6/ D.I. PAGE, D.H. SAUNDERSON, and C.G. WINDSOR, J. Phys. C - 6, 212

/7/E. CLEMENTI, IBM J. Res. Developm. - 9, 2 (1965). /8/ M.L. COHEN and V. HEINE, Solid State Phys. _. 24 (1970). /9/ J, HAFNER, Nuovo Cimento Lettere - 5, 503 (1972); phys. stat. sol. (b) - 56,

- 84, 797 (1977).

(1973).

579 (1973). /10/W.F. KING III and P.H. CUTLER, Phys. Rev. B 5 1303 (1973). /ll/R.S. DAY, F. SUN, P.H. CUTLER, and W.F. KING 111, J. Phys. F - 7,

/lZ/F, SUN, R. DAY, and P.H. CUTLER, Solid State Commun. - 27, 835 (1978).

(Received September 11, 1986)

L169 (1977).