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Vol. 1990. SolidState Communications, 75,No. 4, pp. 345-349, in Printed GreatBritain.

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RESISTANCE SCALING IN MEDIA WITH FRACTAL-LIKE STRUCTUREIN THE VICINITY OF A METAL-INSULATOR TRANSITION I.T. Belash, V.F. Gantmakher O.I. Barkalov, and V.M. Teplinskii Institute SolidStatePhysics, of Academy Sciences the USSR,Chernogolovka, of of Moscowdistrict, 142432, USSR (Received February1990by A.L. Efros) 22 Scalingrelation betweenresidual.t\ and temperature-dependent R, parts of the sampleelectricalresistance the vicinity of a metalin insulatortransitioncan be a guide in examiningthe structureof the material. Three possiblevalues of the exponent v in the relation R, c (Ro)' are discussed. : I is typical for a random mixture of v metallicand insulating domains,v : 0 for granularmetals.A special case v : 0.75 which has been observedrecentlycorresponds a to fractal structureof the insulatingphasewith the classical sizeeffect governing conductivity the metallicchannels. the of Experimental data on Zn-Sb and Al-Ge alloysare presented.

drops to zero. We shall concernourselves only with studying metallicsideof thesample the transition. The TRANSPORT phenomena inhomogeneous in media main ideais that while analyzing scalingrelations for are usually considered applying the percolation by judge what type of the the sampleresistance can one theory []. However,the material science has at its structure realized the sample is in and distinguish, for patterns disposal lot of geometrical a whicharenot of example, fractal pattern from a random one. a granularmetals a simplerandomnature.For instance, contain in essence correlated system of rather , a 2. FRACTAL PATTERN OF THE SAMPLE fbgularly alternatingmetallicand insulatingregions. STRUCTURE An excusefor applying the percolationtheory to granularmetalslies in possible existence random We shall discussan inhomogeneous of structure holesin the intergraininsulatingfilms or in random- createdin the courseof the specialkind of phase nessof thickness thesefilms. However,the exper- transition, namely,the amorphizationof the metaof imentdoesnot qualitatively the percolation fit theory stablephaseof alloys Zn-Sb and Al-Ge quenched as far as the metal volumefraction is considered and [2]. underpressure 6]. The initial stateis crystalline [5, Much more complicatedand interestingstruc- metallicwhile the final one is amorphous and insulatprocess tures may occur when the solid mixture arisesas a ing.The amorphization couldbe led slowlyby process of a phasetran- the low temperature resultof an agigregation or annealing. could be repeatedly It sition in someparts of the sample[3]. If thesetwo interruptedby returningto nitrogentemperatures. phaseshave different conductivities,for instance, This transition to the amorphousstate has the place, one following important features: when a metal-insulatortransition takes l. The specific volume increasessignificantly should describeconducting network and the total conductance the sample. a mediumwith a corre- during the transition[5]. of In lated structurean essential difference from the pre2. The sample resistance increases ordersof R by dictionsof the percolationtheory during the metal- magnitudewhile the sampleremainsmetallicat low insulatortransitionmay be observed It arises from temperatures, while it retainsANAT > 0 [7]. i.e. [4]. correlation the strongspace-time amongmacroscopic 3. The large increaseof the resistance accomis panied a very smalldecrease the temperature volumeswhich undergothe phasetransition. by of d In this content the term "metal-insulatortran- of the superconducting transition.This point contains "point transition"whena the evidencethat the sition" hastwo meanings: sample is macroscopically dielectric,and "sample tran- inhomogeneous the intermediate in states the tranof lsmall volume becomes sition" when the conductance the whole sample sition [7]. of 345

I. INTRODUCTION


346

RESISTANCE scALINc oF A METAL-INSULATOR TRANSITION vot.75.No.4
i Roi/Rol

Y

o (r
t E

r 2 3 4 5 6

3

l 0 t80 450 tooo 2500

(DJ

Zna3 Sb57 (c)

Fig. L Two-dimensional cross-sectionsthesupposed of fractal structureof the amorphousphaseat various Tfi) stages the transition. of The dashed linesare traces of the surfaces which the current pathsare disposed. Fig. 2. at Scaled temperature dependence the Zn_Sb of alloy resistance the sample in states l_5 _ solidline. Basing the first point of this list and on the The samefor the sample the state6 - dashed on in line. resistance data which will be presented below and usingarguments the theory of elasticitya fractatof 3. Zn-Sb DATA like modelof the intrinsicstructure the sample of was Theexperiment theZn-Sballoyandits possproposed [8]. It supposes growinginsulating in with the amorphousinclusionsto be cactus-likewith leaves ible interpretationwere discussed detail in [7, g]. in branchingmany times. Theseleaves,or sheets, briefly the results. are Here we'll first summarize Figure 2 demonstratesthe low-temperature supposed neitherto intersect eachother nor to merge (Fig. l). This maintains existence currentpaths, dependence the sample resistanceduring the of the of amorphization. is drawn with the use of a simple i.e. conductive It channels, the far-gonestages the at of rescaling. For each experimental transition,in contrastwith the percolation curve the residual model. The development the "cactus" structures of & can resistance : R(?" = 7.1K) was subtractedfirst; be described a parameter which hasthe meaning then the ordinatesR(7') - Ro were divided by the by d of the meandistance D between leaves Note that after the and between slopecoefficient = @RIAT)r=rox. the sidebranching two lines.Accordingto the arguments these manipulations: shift of theoriginand the the proposed Esipov[8]thecurrentpathsarelocated .. scaling, the curves all by coincidenot only at two points at the surface with fractaldimension interval. The threewhich separ- but within a rather large temperature atestwo "cactuses"; from the mathematical point of further the amorphousphaseevolvesthe lessis the view the current paths are similar to trajectories dependence universal. is of regionwherethe temperature brownianparticles, random-walktrajectories However,evenwith Ro increased three ordersof i.e. by [9]. The fractal dimension sucha trajectoryis two: its magnitude regionremains the ratherlarge.It is only at of length is proportional to squaredradius K of the the last stageof the transition that the scalingprodomain it occupies. cedure failsto reproduce universal the function(curve 6 in Fig. 2). So, as far aselectrical resistance concerned, is the conductivechannelsare conductivebrownian trajectories. Now d becomesthe step length of the D. (0R/oT)r,rox randomwalk and, with the distance between conthe tacts Z being constant,we have the channellength
). q. d-t.

It is natural to suppose that the cross-section of the channel d2. The simplest is way to introduce this is to usethe latticerepresentation: when the random walk occurson a cubic lattice with a period d the cross-section the celld2is a naturalcross-section of of the trace.Thedevelopment the fractalpatterngiven of on Fig. I corresponds decrease d. This. in turn. to of meansthat instead a latticewith a fixed periodwe of dealwith a series lattices of with graduallydecreasing periods. Hereis, again,a remarkable contrastwith the classical percolation theory.

tg Ro

lig, 3 Double-logarithmicplot of the derivative ARIA! dependence the residualresistance for on rRs two Zn-Sb samples. Fullcircles- dataobtained irom the curvespresentedin Fig. 2. Straight line corresponds the exponent : 0.75. to a


vol. 75,No. 4 RESISTANCE SCALING A METAL-INSULATOR oF TRANSITION
In Fig. 3 the slopecoefficient is plottedagainst D the residualresistance chosento be a measure Ro of amorphization. The experimental results demonstrated Figs.2 in and 3 can be described the relation by R@, r) : Ro(d) + Rt(d)f(T)

347

alongits boundary C-2t3 is d-t. For thewholesystem, whereSC2/3 trajectories parallellyconnected are each beinga chainof ICrl3 linkswith lengthg-ztt4-r un6 with cross-section we shall obtain d-'2.

'. o : Qls)(cd\'

(a)

Here d is the amorphization parameter and d* is its initial value,factorization R(I) - & is a consefor quence the universal of temperature dependence illustratedby Fig. 2, the exponent = 0.75: c D oc R3t'. (2)

Let's represent resistance as a productof a the R p resistivity by a geometrical factor@(thelatterequals Z/S (length/cross section) a homogeneous for cylindrical sample). Then R: p(d, T)$(d) :

Note, that resistivity itselfhassense only at "large whenL > C-tt3.Whent < C-'lt, thedepenscale" p dence on I mustexist.It hasnot beenexperimentally studiedyet. Let us make some numerical estimates.The whole rangeof 'rariation of the resistance observed at the metallicside of the transition(four ordersof magnitude)corresponds tenfold decrease the to of parameter For the sizeeffectto take placethe larger d. limit d* of the interval wherethe thickness varies d (and (l) holds) must be smallerthan the mean free path for bulk material /. The latter can scarcelybe largerthan (l-2) x l0-5cm. So we have

1000A- il 2- d = 100A. lpo@) + p,(d)f(d))Q@). to (3) The lower limit d,;n l00A corresponds curves 5, 6 in Fig. 2. Using a usual estimate the carrier for From (2) and (3) it follows densityin the metallicphaseand the initial apparent valueR(S/L) r 5Og0cm [6]one getsfrom q [po@)]lQ@\-'. p{d) (4) resistivity (3)and(8)thatC1d*1t l-10,i.e.C = 1015-1016crn-3 So not only the factor { but also the resistivity Both estimates, d,1n and C, sound quite reasonable. "knows" about the parameter As d is the width of d. the conductive channels, this may be explained with 4, Al-Ge DATA the help of the classical sizeeffectin the resistance. - the mean free path in the bulk Let's introduce/0 The experimentwith the Al-Ge alloy was permaterialdetermined the scattering point defects formedsimilarlyto thosewith Zn-Sb. The differences by by and the temperature part /, of the bulk wereonly quantitative(seethe Table).However,the dependent meanfreepath i(?n): : [(I)]-' - /t'. The usual resultsare quite different,as it can be seenfrom /;' the expression resistivity for is comparison Figs.4 and 5 with Figs.2 and 3. Note of that Fig. 5 demonstrates exponent : I instead the c prl prll. l\ p: (5) of a = 0.75. Exponent0.75in equation(2) is the resultof the (n is the carrierdensity, is the Fermi momentum). combinationof a fractal structureof the insulating p. For a wire with diameter 4 16 meanfreepath /o phase d the and a ratherlong meanfreepath /. If, instead, can be,within a rathergood accuracy, replaced (5) I 4 d^a holds one will find in by d [0]. Then,with whatever relationbetween and /, R,(d) oc &(d) = [&(d)]'. (9) d. one has

;?7 = ;A\i* i)

po(d) r. d-t ,

p,(d) t

do.

(6)

Combiningthis with (4) we finally obtain:
Q(d) q ddt/g-t) : d-t, (7)

wherethe right exponent corresponds experimental to valuea : 0.75. Equation(7) fits the fractalmodelperfectly.Ifthe concentration initialleafnuclei Ccm-3theneach of is "independent"fractal shownin Fig. I has the space sizeof C-rlr and the lengthof a trajectory situated

Similarity of the processes the both alloys [5] in givesreasonto suggest that the structuresin both are the same and that the relationbetween and d,,nis the / main sourceof the difference. The relation (9) is far less informative than relation(2). It holdsalsoduring a uniform transition when the changes the resistivityare controlledby in the numberof carriersn in (5) as well as in the case whenthe resistance changes due only to the alterare nation in the shape the conductive of channels, of i.e. the factor {. The latter caseapparentlytakesplace


348

RESISTANCE SCALING A METAL-INSULATOR oF TRANSITION vot. 75.No.4

l tg Ro 2

Fig. 5. The sameas in Fig. 3 for a Al-Ge sample.
T (K)

conneeted series in with metalvolumes. thetunnel As resistance the films doesnot dependon I one gets of
R'(d) : const oc [Ro(d)]0,

In Fig. 5 apart from the well definedregionwhere(9) holds there is a region where the data can be interpretedas satisfying (10). Probably,this reflects the final stageof the sampletransitionwhen d becomes sheets merge. Then the duringapproaching transition a randommix- too smalland the insulating the in fractal structuretransforms into the granularone. ture of two phases. Indeed,when approaching the threshold, what altersis the sizeof the cellsof the 5. CONCLUSION conductive backbone the infinitecluster.However, in the links between nodesof this backbonealways the In conclusion, the scalingrelationsbetweenthe contain so calledred bonds,regionswith minimal parts of the resistance the vicinity of the metalin possible cross-sections l2]. So thecontribution of insulator transition contain information about the Il l, the sizeeffectto the link resistivity does not alter macrostructure the sample. of (9) Equations and (10) duringapproaching transitionand only the factot express two limiting cases the the which correspond a to random mixture of phases @changes. and to a granularmetal. Insulating films in an ideal granular metal are The exponenta in (l) can be changed the dc size by effect. The Zn-Sb experimental data give such an example. Tablel. Zn-Sb Initial "apparent"resistivity &'S/1, pocm Rangeof resistance changes at the metallic sideof the transition &/Ro' Shift of the superconducting transition temperature 07,l0 ln Rn,K Temperature which the at sample transition occurs, approximately, K Al-Ge A.cknowledgements Authors acknowledgehelpful discussions S.E.Esipovand E.G. ponyatovikii. with REFERENCES l0o
102 l.

Fig. 4. Temperature variationof the Al-Ge sample resistance differentstages amorphization. at of Each next curve is shifted0.02 upward for claritv. At the right of the curvesthe resistance ratios Ro,/Ro, are indicated.

(10)

50

r00
S. Kirkpatrick,Rev.Mod. Phys.45,S74 (1973).

2. Y. Shapira& G. Deutscher. Phys. Rev.B. 27,
4463(r983). 0.16

0.7*

3. D, Kessler, Koplick & H. Levine,Adv. in J.
4.

190

300

*When Ro/&, became the transition 10, splitted into two demonstrating existence odd phasesin the of intermediate state.

(1988). Phys.37,255 D.S. Mclachlan, So/idState Commun. 69,925 (le8e). 5. I.T. Belash, V.F. Degtyareva, E.G.Ponyatovskii & V.L Rashupkin,Fiz. Tverd. Tela 29, 1788 q987) [Sov.Piys.-SolidState 29,1028(1987)]. 6. O.L Barkalov,LT. Belash, V.F. Degtyareva & E.G. Ponyatovskii,Fiz. Tverd. Tela 29, l97S Phys.-Solid (1287); State29, 1138 (1987)]. [Sou. O.L Barkalov,I.T. Belash, V.F. Gantmakher, E.G. Ponyatovskii V.M. Teplinskii,pis'ma &


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RESISTANCE SCALING OF A METAL-INSULATOR TRANSITION

(Edited J.M. Ziman),Cambr. Electrons by Univ. (1969). Press 8. V.F. Gantmakher, Esipov V.M. Teplin- il. B.I. Shclovskii A.L. Efros.Electronic S.E. & & Pronerskii. Zh. Esksp. Theor.Fiz. 97, 373 (1990); i tiesof DopedSemiconducrors, Springer, Beilin, (The first Russian 1984 editionin 1979). lSov.Phys.JETP 70, No I (1990)1. 9. J. Feder, Fractals,Plenum Press,New York 12. D. Stauffer, Introduction Percolation to Theorv. ( I 988). Taylorand Fransis, London(1985). 10. R.G. Chambers The Phvsics Metals. I. in: of

JETF 48, 561 (1988);UETP Lett. 48, 609,

( re88)1.