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Physica 165&166(1990)311-312 B North-Holland

RLSISTA,I{CE SCAIIN]G IN INHOIiO.}ENEOUSMFDIA IN TI{E VICINITY

OF A il{ETAL-INSIII.ATOR TRANSITION

O.l.Llarkalov, Institute of

LT.Belash, Solid State

S.E.Esipov, lthysj-cs,

V.F.Gantmakher, E.C.Ponyatovskii Sciences of

and V.M.Teplinskii 142432, USSR

AcadernJr of

the USSR, Chernogolovka,

Scaling relation B. and temtrrerature-dependent between residual of the sample {, parts resistance in the vicinity of a metdl-insulator transition can b6 e euide in exarnining ture of the material. In the relation 814(Ro)the exponent value v=l- is typical for

electrical the struc-

a random mix-

ture of metallic domains, v=0 for granuler netals. and insulating A speciel case r=0.75 has been observed recently. It corresponds to a fractal structure of the insui.ating; phase with the classigoverrrir c-af size effect the conductivity of the metallic channels. Experimental data on Zn-Sb and Al Ce alloys are presented.

INIRODUCiION -frmsport phenomena in inhomogeneous media ar-e usually considered by applying the percola(1). However, the material tion theory science has at its disposal pat_ a lot of geometrical terns which are not of a sj_mple random nature, lor iristance, granular metals contaj.n in essence a correlated system of rather regularly alternating metallic and insulating regions. Much more conrplicated structur.es may occur when the solid mixture arises as a result of a phase transition in some parts of the sernple. If these two phases have tlifferent conductivities, for instance, wherr a metal-insulator transition takes place, orie shouid describe conducting network and the total concluctance of the sample. In this content the term "metal-insulator rra"point risitjon" has two meanings: transition" when a small volme becomes dielectric, and "sample transition" r'hen the conductance of the whole sample drops to zero. We shall concern orrselves only with stuilying the metallic side ol the sample transition. the main idea is that whi.le analyzing scaling relations for the sample resistance one can judge what type of the struclure is realized in the sample and distinguish, for example, a fractal pattern from a random one. 2. RESIST.A-\CE SCALING Let's represent the resistance B as a product of a resistivity factor e P by a geonetrical

1.

holds, The latter case apperently takes place during approaching the trarrsition in a rendom mixture of two phases. Indeed, when approaching the threshold. what alters is the size of the cells of the conductive backbone in the infinite cluster, i . e. the factor p . granu]-er netal Insulating films in an ideal are connected irr series with netal volumes. As the tunnel resistance of the films does not
depend on I one gets Rr = "ontt o [Ra]o .

/s/

= = lru ( R=P(r)Q [po+o1(r)]a
ne'

'o

+11)a 't
free

/r/
path

(p. I

is the -=lo -+11 -

Fermi

momentum,

mean

).When the chenges in the resistivity

are controlled by the carrier density n as well as when the resistance changes are due only to the alternation in the shape of the conductive channels, i.e, of the factor t0, then reletion

BroEo=[Ro]1

/2/

FRACTAL PATTERN OF THE Zn-Sb ALLOY S1RUCTURE Tfe exponent values /=1 from Eq,./2/ and v=Q from. Eq. /3/ are not the only possible ones. Be:'lov follows an exarnple with v=0.75: an inhomogeneoud structure created in the course of a special phase transition, nmely, the amorphization phase of alloy of the metastable Zn-Sb quenched under pressure (2,3), The initial state is crystalline and metallic while the final one is amorphous and insulating. proThe anorphization cess could. be led slowly by the 1ow temperature anneeling. It could be repeatedly interrupted by returning to nitrogen temperatures. This transition to the amorphous stgte has the following important features: 1. the specific volume increa,ses sienificantly during the transition ; 2. the sample resistance .B increases by orders of magnitude while the sample rennins metallic at low temperatures, i . e. while it dA/OnO: retains 3. the large increase of the resistmce is accomparried by a very small decrease of the temperature of the superconductirrg trarrsition. I This point-contains the evidence that the sample ls rnacroscopically inhonogeneous in the interme(2) diate states of the transition A fractal-like model of the intrinsic structure of the sarnple was proposed in (3), It supposes the growing insulating amorphous inclu-

3.

092I-45261901$03.50 1990- ElsevierScience Publishers B.V. (North-Holland) @


3t2

O.L Barkilov, I.T. Belash,S.E. Esipov, V.F. Gantmakher,E.G. Ponyatovskii, V.M. Teplinskii

sions to be cactus-like with leaves brmchins man3r times. These leaves, or sheets, are supposed neither to intersect each other nor to merge. This maintains paths, the existence of current at the far-gone stages of the trensition, rn contrast with the percolation model. The development of the "cactus" structures can be described by a psrameter d, mean distance between the leaves. According to (3),the current paths are located at the surface with fractal dimension three which separates two "cactuses"; point from the mathematical of view the current paths are similar to trajectories of brownian particles. The fractal dimension of slrch a tr.ajectory is two: its Ierigth is proportional to squared radius K of the domain it occupies. So, as far as electricel resistance is concerned, the conductive channels are conductiwe brovrnian trajectories. Now d becomes the step length of the randorn walk and, with the distance between the contacts I beins constant, we have the chmnel Iength h"rd-' . Supposing in addj-tion thet the cre.ss-section of the channel is d- we -. obtain t!{d To get an exponent in Eq. /2/ different from u=1 we need some dependence of t0 on d, Such dependence can exist due to the dc size effect. For a wire with diarneter cK -a

leD

I

lg

Ro

FICL'RE 2

get
7F

Bo,d

'

-a

R.od

-

,

R,lo[q, j*

n

"

/5/

That is just what follows fr"om the experiment... (Fie.1). Note, in passir, that in the percoJ.-ative system the ]inhs between the nodes of its backbone always contain so called red bonds, regions with minirnal possible (1). cross-sections This means that. the dc size ef f ect cmot influence the - value.

4. Al -Ge DATA The experiment with the Al-Ge alloy was performed simllarly to those with Zn*Sb. The differen ces were only quurtitative"However, the results are quite different, as it can be seen from the comparison of Figs.1 and 2. Note that Fig.2 demonstrates the exponent r)=1 instead of ,=0.2S. Similarity of the processes in the both alloys gives reason to suggest that the structur"es in both are the same and that the reletion berween I and d is the main source of the difference. Exponent 0.75 is the result of the combina, tion of a frectal structure of the insulating phase and a rather long mean free path J. If, ins1.ead, I <.i d holds one will find y=1. II)NCLTISIONS in conclusion, the scaling relations between the pafts of the resistance in the vicinity of the metal-insulator transition contain informa_ tion ebout the macrostructure of the sample. Eqs /2/ xd /3/ express the two liniting cases which correspond to a rarrdom mixture of,phases and to a gramular metal, The exponent D can be changed by the dc size effect. lhe Zn-Sb experimental ciata give such an example. REFM.E\]CES { 1 ) D. Stauffer . Introduction percolation to theor.y (Taylor and Fransis, London, 19g5)O.I.Barkalov, \2) LT.Belash, V.F.Gantrnakher, E.G.Porwatovskii and V.M.Teplinskii, pis'me JETF 48 (1988) 561 [JE-II,Letters 48 (1988) 609.1 . ( 3 ) V.F.Gantnakher, S.E.Esipov md V.M.Teplinskii. Zh, Ehsp. i Theor. Fiz. 92 (1990) 373 [Sov. Ptrys. JETP 70, No 1 (1990)]. (4) J.Feder. Fractals (Plenum press, New york, 1988). R.G.Chambers in : The Physics of Metals_ \c/ l.Electrons, ed. J.M.Ziman, (Cambr. Univ. Press, 1969).

o _ 16p70T)r_366

h0

ls Ro FIGURE 1