On the temperature dependence of the electrical resistivity of non-stoichiometric actinide compounds

Solid State Communications, Vol. 27, pp. 931-932. © Pergamon Press Ltd. 1978. Printed in Great Britain.

0038-1098/78/090 i -0931 $02.00/0

ON THE TEMPERATURE DEPENDENCE OF THE ELECTRICAL RESISTIVITY OF NON-STOCHIOMETRIC ACTINIDE COMPOUNDS F. Brouers Institut fiir Theoretische Physik, Freie Universit~t, Berlin 33, West Germany and A.A. Gomes, O.L.T. de Menezes and A. Troper Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil (Received 31 May 1978 by S. Amelinckx)

We suggest that the systematic investigation of the dependence of resistivity upon temperature in non-stochiometric actinide compounds could bring some information on the possibility of non-Kondo minima in highly resistive systems suggested by a recent theory of Markowitz. RECENTLY Markowitz [1 ] has shown that the inclusion of the temperature dependent Debye-Waller factor in the calculation of the residual resistivity can lead to a resistance minimum at low temperature and to a resistance maximum at high temperature when the residual resistivity is large enough. Quite generally the equation of resistivity can be written in the form

Boltzmann constant. The bar over Wx means that average over K has been taken. The ideal resistivity [3] (e z -- 1)(1 - - e -z)

Pi(T) = Pl

(6)

gives for T'< 0 ,Oi(T)

"~"

124.4 pl ( T ) s

(7)

p(T) = [Po + pi(T)] e -2wiT) [Po + p~(T)] [ 1 - 2 W ( T ) ]

(1)

Po is the T-independent resistivity resulting from the multiple scattering of electrons by random defects, pi(r) is the ideal phonon resistivity due to real single phonon emission and absorption. The exponential factor is the Debye-Waller factor which corresponds to the infinite series of coherent creation and destruction of virtual phonons and which acts to decrease the scattering rate and is given by Ziman [2]

and for T >> 0

Therefore at low T, the resistivity is given by

o(7) po 1 -?-w-3290g, + 124.4p ~

(9)

and at high T by

p(7) At low temperature T ~ 0 where 0 is the Debye temperature W(T) "" 1.645W1

+--~--.

(3)

po

+(o, -2 po)

T

(10)

The first expression gives rise to a maximum at tempera. ture T -- "" 0.2WV3l'--z-~I

o

At high temperature T >> 0

(11)

\p,]

while the relative depth of the minimum is given by 4

(4) [ P ( 0 ) - - p m , ] [p(0)]-' = W~3/P°] ~3

\o,]

The factor WI(K) = 3h2K2/2MkBO is defined in terms of the constant h, M, the mass of the electron, k n the 931

(12)

932

TEMPERATURE DEPENDENCE OF NON-STOCHIOMETRIC ACTINIDE COMPOUNDS Vol. 27, No. 9

Markowtiz theory has been derived for a model of nearly free electrons. For disordered metallic systems containing transition of rare-earth components an investigation of the coupling between thermal and configuration disorder has been derived recently. The resistivity at high T and the possible occurence of a maximum has been discussed. Using a tight-binding picture and the theory of disordered concentrated alloys one can show (Harris et al. [4] ) that if' has to be multiplied by a factor depending on the characteristics of the band. At high temperature an expression similar to (10) has been obtained: T

p(T) ~ Po + (P, --N(E£, V)Wpo)-~

of UCo2 reported by Hrebik and Coles [6] could be qualitatively be interpreted using formula (11) and (12) since Tmi~ as well as the depth of the minimum increase with P0In these systems however one has to include a spinfluctuation contribution which at low T behaves as A (T/T~f) 2 where Tsf is the spin-fluctuation temperature: It is therefore interesting to consider the effect of such a term in the framework of Markowitz theory. If one includes this contribution, the equation giving the minimum temperature is obtained by solving the third order equation

(13)

where N depends on the scattering potential and the position of the Fermi level. In a two hybridized s-d (or d - f ) two-band model when the Fermi level is in a narrow peak of the density of states, N(EF, 11) is proportional to the square of the narrow-band density of states and to a factor {1 -- 3 [(EF -- V)/F] 2 } where P is the width of the narrow-band peak, and can be positive or negative according to the position of the Fermi level (Brouers [5]). Therefore in systems containing narrow bands, Markowitz theory should keep its relevance provided W is multiplied by a factor which can be large when the Fermi level lies in a narrow density of states peaks. At low temperature, resistivity minima are generally interpreted as Kondo resistance minima or "Kondo-like" resistivity minima. However it is interesting to examine these data in the light of Markowitz mechanisms. For instance the experimental data on the influence of nonstochiometry on the resistivity and magnetic properties

[ 0 ~2_ [T~2

-- 6.58W00 + A

= 0

and will depend on the ratio of the Debye and the spinfluctuation temperatures and of the coefficient A which can vary largely from one intermetallic actinide system to another (Iglesias-Sicardi et al. [7]). I f A is small its effect is simply to decrease the temperature of the minimum:

(~-)

3=6"58~p°-A(O/Tsf)2622pl

We believe that a systematic investigation of stochiometric disorder in intermetallic actinide compounds with or without spin fluctuation contributions could be a way to prove or disprove the possibility of non-Kondo minima in these systems.

REFERENCES

D.,Phys. Rev. B15, 3617 (1977).

1.

MARKOWITZ

2.

ZIMAN J.M., Electrons and Phonons, p. 364. Clarendon Press, Oxford (1960).

3.

ZIMAN

4.

HARRIS R., SHALMON M. & ZUCKERMANN M. (to be published).

5.

BROUERS F., J.

J.M.,Principles of the Theory of Solids, p. 62. Cambridge University Press (1971).

6.

de Phys Lett. (to be published). HREBIK J. & COLES B.R., Physica 86-88B, 169 (1977).

7.

IGLESIAS-SICARDI J.R., GOMES A.A., JULLIEN R. & COQBLIN B.,J.

Low Temp. Phys. 25, 47 (1976).