Short Notes K73

phys. stat. sol. (b) - 133, K73 (1986)

Subject classification: 72.15; S1.2 Department of Physics, IndianInstitute of Technology, Delhi The Width of the Scattering Resonance and the Electrical Transport Properties in Partially Filled d-Band Transition Metals

1)

BY SADHANA GARG, H. C. GUPTA, and B. B. TRIPATHI

Recently, transport properties of liquid transition metals and alloys are actively investigated, both from experimental and theoretical view point /1 to 4/. Among these properties the electrical resistivity of the partially filled d-band transition metals and their alloys is of special interest for the theoreticians because till date none of the theories were really able to explain it, One of the reasons is the fact, that these transition metals have a very high resistivity a t their melting point,

The electrical resistivity of simple liquid metals can be described by Ziman' s theory /5/. This formulation was extended to liquid transition metals and their alloys /6, 7/, where d-electrons a r e treated as resonance states scattered by muffin-tin potentials in the conduction band. Evans et al. /6/ as- sumed that the d-phase shift dominates a t the Fermi energy in transition metals. Therefore they have expressed the resistivity, because of the resonant nature and large magnitude of the 1 = 2 phase shift as

All the terms used in the above expression have the same meaning as given by Evans et al. /6/. The resistivity, 8 , depends upon Ed and width r of the scattering resonance, because

The term involving in the above expression corresponds to N(EF) in the Lloyd single site approximation to the density of d-states /B/. Further it has

1) Hauz Khas, New Delhi 110016, India.

K74 physica status solidi (b) 133

‘i

4 18.7 13.34 0.85 7.76 67 6.40

been observed that the density of states per atom may be a reasonably universal function of the electron to atom ratio within a given structure /9/. Therefore, in the present treatment, we have considered the N(E ) dependence upon the valency.

F

In order to investigate this dependence, we find that the choice of the par-

ameters Z, EF, Ed, and l? used is very important in calculating the electrical transport properties. We have used an approach in which we have initially estimated I’ , using the resistivity as input parameter for different valency of partially filled d-band transition metals such as Ni ( Z = 2), Ti (Z = 4), V( Z = 5), and W (Z = 6), the numerical calculations having been made using expression (3) where k F = 3 ?t z/sZo and EF = k F h /2 m.

of liquid metals by Waseda /lo/ and Ed is taken from the work of Harrison /9/.

The calculated values of r a s shown in Table 1, suggest that r is the multiple of a constant which has a value equal to %/8. Hence a relation for r is pro- posed as

3 2 2 2

The liquid structure factor a(2 has been taken from the neutron studies %)

(4) Z r = 2 ?r/8 eV ,

where Z is the valency. The width of the scattering resonance (r ) obtained from (4) suggests in transition metals I? to be a rather universal function of valency. With this expression of r , the electrical resistivity is calculated for

V

5 22.0 18.44 0. 96 8.13

135 12.46

T a b l e 1

Data used for calculating the width of the scattering resonance I?

system N i

2 17.6 11.76 1.18 6.91 85 1.59

N

6 22.4

19.16 0. 8l*) 9.45

35 25.45

*) Using the a ( 2 5 ) exp data of C r scaled to the W value of 2 %, because the structure factor for W has not yet been measured.

Ta

bl

e

2

Res

ults

of

resi

stiv

ity

calc

ulat

ion

for

liqui

d tr

ansi

tion

met

als

Ni 2

2

17.6

11

.76

6.91

82

85

b)

60

104 55

syst

em

Pd

15.9

9.

64

4.52

84

83

b)

- - 58

-

?e 2 17.1

11

.u

7.64

-

.37

.4 Ob

) 18

2

183 82

:0 2 17

.5

11.6

4 7.

36

91

oob)

28

100 83

4 18.7

13

.34

7.76

64

67

‘)

89

34

.

56

V

5 22.0

18

.44

8.13

13

5 13

5‘)

3 87

794

Z81

W

6 22.4

19.1

6 9.

45

135

135‘

)

Mo 6 22

.5

19.2

8 7.

12

1244

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ata

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/9/

, b)

Res

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c) R

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/3/,

d) R

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rom

/l/

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Res

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fro

m /2

/, f)

Re-

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from

/4/.

9 T

he d

isag

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ent b

etw

een

theo

ry a

nd e

xper

imen

t ca

n be

att

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to th

e va

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of

a(2k

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aken

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calc

ulat

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from

the

tab

le o

f C

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ased

a /lo

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the

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a

(25

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Cr 6 24

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1

818

120

physica status solidi (b) 133 K 76

different partially filled d-band transition metals using (3) ,the resul ts are shown

in Table 2 . Gallagher and Greig /2/ have discussed the Ziman formalism and found that i t is difficult to see how any reasonable choice of EF, kF , and Z could bring the

values of Ziman’s formalism in line with experimental resul ts , where one is tempted

to use Z>1 for the transition metals. The present investigations deal with the actual valency of Z for most of t h e transition metals, thereby modifying the value of EF and % as compared to previous calculations /4/, with Z = 1, but we observe that periment. The values of Ed are the s a m e as used by Gallagher and Greig /2/

and now i f we see Table 2 for the excellent agreement obtained between theory and experiment, the reason only is that the expression for the value of the width of the scattering resonance I? , which we have proposed as (4), is aunwersal lunc- tion of valency.

In conclusion, we can say that the present formalism for r gives excellent

= 1 does not give a good agreement between theory and ex-

agreement in resist ivity for most of the partially filled d-band transition metals.

One of us (S. G.) is grateful to CSIR, India, for financial assistance.

References

/1/ J .B. VAN ZYTVELD, J. non-crystall. Solids - 61/62, 1085 (1984).

/2/ B. L. GALLAGHER and D. GREIG, J. Phys. F - 12, 1721 (1982). /3/ V. SEYDEL and W. FUCKE, J . Phys. F 10, - L 203 (1980). / 4 /K. HIRATA, Y. WASEDA, A. JAIN, a n d R . SHRIVASTAVA, 3. Phys. - 7 ,

/ 5 / J.M. ZIMAN, PlectronsandPhonons, ClarendonPress, 1960 (Chap. 5).

/6/ R. EVANS, D.A. GWENWOOD, a n d P . ljLOYD, P h y s . L e t t e r s ~ 35, 57 (1971).

/7/ 0. DREIRACH, P. EVANS, H. J. GUNTHERODT, and H. U. KUNZI,

/8 /P . LLOYD, Proc. Phys. Soc. 90, - 207 (1967). /9/ W.A. HARRISON, Electronic Structure and the P rope r t i e s of Solids,

Freeman, San Francisco 1980 (p. 491).

419 (1977).

-

J. Phys. F - 2, 709 (1972).

/lo/ Y . WASEDA, The Structure of Non-Crystalline Materials, McGraw-Hill, 1980.

/ll/ N.F. MOTT, Phil. Mag. 26, 249 (1972). - (Received October 10, 1985)