review - granular electronic systems

8/2/2019 Review - Granular Electronic Systems

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Granular electronic systems

I. S. Beloborodov, A. V. Lopatin, and V. M. Vinokur

Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

K. B. Efetov

Theoretische Physik III, Ruhr-Universitt Bochum, D-44780 Bochum, Germany

and L. D. Landau Institute for Theoretical Physics, 117940 Moscow, Russia

Published 2 April 2007

Granular metals are arrays of metallic particles of a size ranging usually from a few to hundreds ofnanometers embedded into an insulating matrix. Metallic granules are often viewed as artificial atoms.Accordingly, granular arrays can be treated as artificial solids with programmable electronicproperties. The ease of adjusting electronic properties of granular metals assures them an importantrole for nanotechnological applications and makes them most suitable for fundamental studies ofdisordered solids. This review discusses recent theoretical advances in the study of granular metals,emphasizing the interplay of disorder, quantum effects, fluctuations, and effects of confinement. Thesekey elements are quantified by the tunneling conductance between granules g, the charging energy ofa single granule Ec, the mean level spacing within a granule , and the mean electronic lifetime withinthe granule /g. By tuning the coupling between granules the system can be made either a goodmetal for ggc =1/2dlnEc / d is the system dimensionality, or an insulator for ggc. The

metallic phase in its turn is governed by the characteristic energy

=g: at high temperatures T the resistivity exhibits universal logarithmic temperature behavior specific to granular materials,while at T the transport properties are those generic for all disordered metals. In the insulatorphase the transport exhibits a variety of activation behaviors including the long-puzzlingexpT0 /T1/2 hopping conductivity. Superconductivity adds to the richness of the observedphases via one more energy parameter . Using a wide range of recently developed theoreticalapproaches, it is possible to obtain a detailed understanding of the electronic transport andthermodynamic properties of granular materials, as is required for their applications.

DOI: 10.1103/RevModPhys.79.469 PACS numbers: 73.23.Hk, 71.30.h, 73.22.f

CONTENTS

I. Introduction 470

A. Why are granular electronic systems interesting? 470

B. Physical quantities characterizing granular materials 472

II. Normal Granule Arrays 473

A. Transport properties 473

1. Classical conductivity 474

2. Metallic regime 474

3. Insulating regime 477

B. Model and main theoretical tools 478

1. Hamiltonian 479

2. Diagrammatic technique for granular metals 481

3. Coulomb interaction and gauge

transformation 482

4. Ambegaokar-Eckern-Schn functional 483C. Metallic properties of granular arrays at not very

low temperatures 484

1. Perturbation theory 484

2. Renormalization group 485

D. Metallic properties of granular arrays at low

temperatures 486

E. Universal description of granular materials 488

F. Insulating properties of granular metals: Periodic

model 489

1. Activation conductivity behavior 490

2. Mott gap at small tunneling conductances 490

3. Mott gap at large tunneling conductances 4914. Metal-insulator transition in periodic

granular arrays 491

G. Hopping conductivity in granular materials 4921. Density of states 4922. Hopping via elastic co-tunneling 4933. Hopping via inelastic co-tunneling 494

III. Arrays Composed of Superconducting Grains 495A. General properties of granular superconductors 495

1. Single grain 4952. Macroscopic superconductivity 4963. Granular superconductor in a magnetic field 4994. Transport properties of granular

superconductors 499B. Phase diagram of granular superconductors 501

1. Phase functional for granular

superconductors 5012. Mean-field approximation 502

C. Upper critical field of a granular superconductor 5031. Critical field of a single grain 5032. Critical magnetic field of a granular sample 504

D. Suppression of the superconducting criticaltemperature 505

E. Magnetoresistance of granular superconductors 507IV. Discussion of the Results 510

A. Quantitative comparison with experiments 5101. Logarithmic temperature dependence

of the conductivity 510

REVIEWS OF MODERN PHYSICS, VOLUME 79, APRILJUNE 2007

0034-6861/2007/792/46950 2007 The American Physical Society469
http://dx.doi.org/10.1103/RevModPhys.79.469http://dx.doi.org/10.1103/RevModPhys.79.469


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2. Hopping conductivity 5113. Negative magnetoresistance 511

B. Outlook 512Acknowledgments 513Appendix A: Calculation of the Tunneling Probability Pel inthe Elastic Regime, Eq. 2.105 513Appendix B: Calculation of the Tunneling Probability Pin inthe Inelastic Regime, Eq. 2.114 515References 516

I. INTRODUCTION

A. Why are granular electronic systems interesting?

Granular conductors form a new class of artificial ma-terials with tunable electronic properties controlled atthe nanoscale and composed of close-packed granulesvarying in size from a few to hundreds of nanometersoften referred to as nanocrystals. The granules arelarge enough to possess a distinct electronic structure,but sufficiently small to be mesoscopic in nature andexhibit effects of quantized electronic levels of confined

electrons. Granular conductors combine the uniqueproperties of individual and collective properties ofcoupled nanocrystals opening a new route for potentialnovel electronic, optical, and optoelectronic applica-tions. Applications range from light-emitting devices tophotovoltaic cells and biosensors. The intense interest inthem is motivated not only by their important techno-logical promise but by the appeal of dealing with anexperimentally accessible model system that is governedby tunable cooperative effects of disorder, electron cor-relations, and quantum phenomena Murray et al., 1993;Gaponenko, 1998; Mowbray and Skolnick, 2005.

Among traditional methods of preparation of such

materials, the most common are thermal evaporationand sputtering techniques. During those processes me-tallic and insulating components are simultaneouslyevaporated or sputtered onto a substrate. Diffusion ofthe metallic component leads to the formation of smallmetallic grains, usually 350 nm in diameter see Fig. 1.Variations of grain size within a sample produced bythese methods can be kept as low as about 10%Gerber et al., 1997. Depending on the materials usedfor the preparation, one can obtain magnetic systems,superconductors, insulators, etc.

Recent years have seen remarkable progress in thedesign of granular conductors with controllable struc-

ture parameters. Granules can be capped with organic

ligands or inorganic molecules which connect andregulate the coupling between them. By altering the sizeand shape of granules one can regulate quantum con-finement effects. In particular, by tuning the microscopicparameters, one can create the granular materials vary-ing from relatively good metals to pronounced insulatorsas a function of the strength of electron tunneling cou-plings between conducting grains. This makes granularconductors a perfect exemplary system for studying themetal-insulator transition and related phenomena.

One emerging technique to create granular systems isthrough self-assembling colloidal nanocrystals Collier etal., 1998; Murray et al., 2000. For instance, Lin et al.2001 described almost perfectly periodic two-dimensional arrays of monodisperse gold nanoparticleswith grain size variations within 5%, covered byligand molecules that play the role of an insulating layer.Such samples are produced via self-assembling of goldnanoparticles at the liquid-air interface during theevaporation of a colloidal droplet Narayanan et al.,2004. By changing the experimental conditions, multi-layers of nanocrystals can also be created see Figs. 2and 3 Parthasarathy et al., 2001, 2004; Tran et al., 2005.Other important examples include Langmuir films ofcolloidal Ag Collier et al., 1997; Du et al., 2002 andarrays of semiconductor quantum dots Murray et al.,1993; Du et al., 2002; Wehrenberg et al., 2002; Yu et al.,2004.

Of the range of techniques developed for the fabrica-tion of semiconductor quantum dot arrays, the most suc-cessful appears to be the self-assembled technique ofepitaxial growth of two semiconductors having signifi-cantly different lattice constants. For the prototype sys-tem of InAs on GaAs, where the lattice mismatch is 7%,

InAs initially deposited on GaAs grows as a strainedtwo-dimensional layer referred to as the wetting layerMowbray and Skolnick, 2005. These materials andIn2O3 :Sn known as indium tin oxide ITO are themost widely studied systems. They exhibit a high visibletransparency and a good electrical conductance. Theyare used as electrodes in light-emitting diodes Kim etal., 1998; Zhu et al., 2000, solar cells Gordon, 2000,smart windows, and flat panel displays.

All these experimental achievements and technologi-cal prospects call for a comprehensive theory able toprovide a quantitative description of the transport andthermodynamic properties of granular conductors,

which can therefore serve as a basis for the clever design

FIG. 1. Scanning electron microscope photo-graphs of indium evaporated onto SiO2 atroom temperature. From Yu et al., 1991.

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B. Physical quantities characterizing granular materials

This review deals with systems that we call granularmetals, which are well modeled by an array of meso-scopically different metallic particles identical in sizeand shape, with the intergranular electron coupling be-ing described via the tunneling matrix. The grain ar-rangements may be either periodic or irregular.

The effect of irregularities in the grain positions andin the strengths of the tunneling coupling on the physicalproperties of granular systems is different for metallicand insulating samples. If the coupling between thegrains is sufficiently strong and the system is well con-ducting, the irregularities are not very important. In con-trast, irregularities become crucial in the limit of lowcoupling where the system is an insulator. As a matter ofpractice, recent advances in fabrication techniques nowpermit very regular arrays with fluctuations in the gran-ule size within 5% precision Yu et al., 2004; Tran et al.,2005. As regular arrays promise many technological ap-plications further progress in making perfect systems isexpected.

At the same time, disorder related to internal defectsinside and/or on the surface of the individual granulesand to charged impurities in the insulating substrate ormatrix is unavoidable. Even in metallic grains with meanfree paths exceeding their size, electrons that move bal-listically within the granules still scatter at the grain-boundary irregularities; the electron motion becomeschaotic with the resulting effect equivalent to the actionof intragranular disorder see, e.g., Efetov 1997. Thissurface chaotization would be absent in atomically per-fect spherical or, say, cubical granules. In such idealgranules the additional degeneracies of the energy levelswould lead to singularities in physical quantities. How-

ever, the slightest deviations even of the order of 1 orthe order of the electron wavelength of the shape of thegrains from ideal spheres or cubes will lift the accidentaldegeneracy of the levels this would be equivalent toadding internal disorder. Thus one can justly assumethat the grains are always microscopically irregular.Moreover, the coupling between grains is the source ofadditional irregularities. We thus can adopt a model withdiffusive electron motion within each grain without anyloss of generality. In this model the use of Eq. 1.3 forthe mean level spacing is fully justified.

The key parameter that determines most of the physi-cal properties of the granular array is the average tun-

neling conductance G between neighboring grains. It isconvenient to introduce the dimensionless conductanceg corresponding to one spin component measured inunits of the quantum conductance e2 / : g = G / 2e2 / .As we will see below, samples with g1 exhibit metallictransport properties, while those with g1 show insulat-ing behavior.

One of the most important energy parameters of thegranular system is the single-grain Coulomb chargingenergy Ec. This energy is equal to the change in energyof the grain when adding or removing one electron, andit plays a crucial role in the transport properties in the

insulating regime when electrons are localized in thegrains, such that the charge of each grain is quantized.The physics of the insulating state is closely related tothe well-known phenomenon of the Coulomb blockadeof a single grain connected to a metallic reservoir.

The behavior of a single grain in contact with a reser-voir has been discussed in many articles and reviewssee, e.g., Averin and Likharev 1991, Averin and Naz-arov 1992, and Aleiner et al. 2002. The main features

of the Coulomb blockade can be summarized as follows:i If the grain is weakly coupled to the metallic contact

g1, the charge on the grain is almost quantized; this isthe regime of the so-called Coulomb blockade. ii In theopposite limit g1, the effects of the charge quantiza-tion are negligible and electrons are freely exchangedbetween the granule and reservoir.

Although the systems we consider are arrays of inter-connected granules rather than a single grain coupled toa bulk metal, a somewhat similar behavior is expected.In the regime of strong coupling between grains g1,electrons propagate easily through the granular sampleand the Coulomb interaction is screened. In the opposite

limit of low coupling g1, the charge on each grainbecomes quantized as in standard Coulomb blockadebehavior. In this case an electron has to overcome anelectrostatic barrier of the order Ec in order to hop ontoa neighboring granule. This impedes the transport attemperatures T lower than Ec.

Throughout this review we always assume that themean level spacing from Eq. 1.3 is the smallest en-ergy scale. In particular, in all cases we assume that thecondition Ec is satisfied. This is the most realisticcondition when dealing with metallic particles of a na-nometer size scale given that the charging energy Ec isinversely proportional to the radius a of the grains,

whereas the mean level spacing is inversely propor-tional to the volume V and also taking into account thehigh density of energy states in metals. Note that thismay not be the case in arrays of semiconductor dots,where Ec and may appear of the same order, but thisgoes beyond the scope of our review.

Another important thing to remember is that the in-tergranular tunneling conductance g is much smallerthan the intragrain conductance g0 by the very meaningof the notion of a granular system:

gg0. 1.4

The intragrain conductance g0 is caused by scattering onimpurities or on the boundaries of the grains and theinequality 1.4 means that the grains are not very dirty.The case g g0 can be viewed as a homogeneously dis-ordered system and we dwell on this limit here. Thesingle-grain conductance g0 can be most easily definedas the physical conductance of a granule with cubic ge-ometry measured in units of the quantum conductancee2 / h. In mesoscopic physics it is customary, however, torelate the single-grain conductance g0 to the single-grainThouless energy ETh as g0= ETh/, where the energy EThis defined as




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ETh = D0/a2, 1.5

and D0= vF2/ d is the classical diffusion coefficient. Other

parameters are vF, the Fermi velocity, , the elastic scat-tering time within the grains, and d, the dimensionalityof the grain. The length a in Eq. 1.5 is the linear sizeradius of the grain. If the grains are not very dirty, suchthat electrons move inside the grains ballistically, the

mean free path l= vF should be replaced by the size ofthe grains 2a. The Thouless energy ETh, Eq. 1.5, is pro-portional to the inverse time that it takes for an electronto traverse the grain. The energy ETh exceeds the meanlevel spacing , Eq. 1.3, and therefore the intragrainconductance g0 is always large g01, whereas the inter-granular conductance g can be either larger or smallerthan unity.

The above parameters make up a full set of variablesdescribing the properties of normal granular metals. Ifthe constituent particles are made of a superconductormaterial, a wealth of new interesting phenomena arises.The behavior of such a system can be quantified by add-

ing one more energy parameter, the superconductinggap of the material of a single granule. Now, if theintergrain coupling is sufficiently strong, the system canturn into a superconductor at sufficiently low tempera-tures. The properties of such a superconductor are notvery different from those of bulk superconductors.

On the contrary, in the opposite limit of weak cou-pling between the granules, an array of superconductinggrains can transform into an insulator at T=0. In thisregime Cooper pairs are locally formed in each grain butremain localized inside the grains due to the strong on-site Coulomb repulsion leading to the Coulomb block-ade. Because of localization, the number of Cooper

pairs in each grain is fixed and, according to the uncer-tainty principle, this leads to strong phase fluctuations.Thus the system does not develop global coherence, andglobal macroscopic superconductivity is suppressed.One can describe this effect using the model of super-conductor grains coupled via Josephson junctions. Sucha system is characterized by three energy parameters:the superconductor gap , characterizing a single grain,the Josephson coupling J, and the grain charging Cou-lomb energy Ec. Strong Josephson coupling JEc sup-presses the phase fluctuations leading to the globally co-herent superconducting state at sufficiently lowtemperatures. IfJEc, the Coulomb blockade prevails,

the Cooper pairs become localized at T0, and the sys-tem falls into an insulating state.Note that even in the insulating state the supercon-

ducting gap still exists in each grain and its value isclose to the gap magnitude in the bulk, provided .If the latter inequality is not satisfied, the superconduct-ing gap in the grains is suppressed or can even be fullydestroyed Anderson, 1959, 1964. There are interestingeffects in this regime but we do not consider them herebecause the relevant region of the parameters corre-sponds to either too small grains or too weak supercon-ductors.

At low temperatures, the Josephson coupling J is ex-pressed via the tunneling conductance g as J=g / 2Ambegaokar and Baratoff, 1963 and, at first glance,one might conclude that the transition between the in-sulating and superconducting states should have hap-pened at g Ec /. However, this simple estimate holdsonly in the weak-coupling regime g1, which assumesEc. At stronger coupling, the Coulomb energy Ec is

renormalized down to the value EcE /g Larkin

and Ovchinnikov, 1983; Chakravarty et al., 1987 due toelectron tunneling between neighboring grains. There-fore in the limit of strong coupling g1 the effective

Coulomb energy E is always smaller than the Josephson

coupling EJ, implying a superconducting ground state.Shown in Fig. 4 is the schematic phase diagram summa-rizing the above considerations Chakravarty et al.,1987. For many superconducting granular samplesavailable experimentally the ratio Ec / is large. In thiscase, as one can see from the phase diagram in Fig. 4,the transition between the superconducting and insulat-ing phases at T0 occurs at g 1 Orr et al., 1986;Chakravarty et al., 1987.

To conclude our brief introduction to granular super-conductors, we note that, for most of the experimentallyavailable samples, the grains are much smaller than thebulk superconducting coherence length of the granulematerial,

a 0. 1.6

This allows one to neglect variations of the supercon-ducting order parameter inside the grains and treat asingle grain as a zero-dimensional object. As the condi-tion 1.6 is at this point the most common experimentalsituation, we further concentrate mainly on this regime.

As we now see, depending on the parameters of the

system, many different physical situations appear, andthe phase diagram of a granular material is quite rich. InSecs. II and III we focus on systems consisting of normaland superconducting grains, respectively.

II. NORMAL GRANULE ARRAYS

A. Transport properties

We start our discussion with the properties of granularsystems consisting of normal grains by presenting themain results and their qualitative explanations. This may

FIG. 4. Schematic phase diagram for granular superconductors

at temperature T=0. Symbols S and I stand for superconduct-ing and insulating phases, respectively.

473Beloborodov et al.: Granular electronic systems



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help the reader to understand the basic physics of thesystem and to learn important formulas suitable for adirect comparison with experiments without going intothe technical detail of theoretical models, which will bepresented in subsequent parts of each section.

1. Classical conductivity

The key parameter that determines the transport

properties of granular materials is the tunneling conduc-tance g. In the strong-coupling regime g1 a granulararray has metallic properties, while in the opposite case

g1 the array is an insulator. The insulating state ap-pears as a result of the strong Coulomb correlations thatblock electron transport at low temperatures. Generallyspeaking, apart from the Coulomb interaction effectsone also has to consider the effects of quantum interfer-ence that may play an important role in low-conductingsamples. In homogeneously disordered systems interfer-ence effects play an important role, leading to the local-ization of electron states in the absence of interactionAnderson, 1958.

In the metallic regime and at high enough tempera-tures, both Coulomb correlation and interference effectsare weak. In this case the global sample conductivity 0is given by the classical Drude formula; in particular, fora periodic cubic granular array

0 = 2e2ga2d, 2.1

where a is the size of the grains and d is the dimension-ality of the sample. Equation 2.1 has a straightforwardmeaning: in order to obtain the physical conductance ofthe contact, one should multiply its dimensionless con-ductance g by 2e2 the factor 2 is due to spin and =1and, then, multiplying the result by a2d, one arrives at

the conductivity per unit volume.Although Eq. 2.1 is written for a periodic array, theelectron system is not translationally invariant, other-wise the sample conductivity would be infinite. Equation2.1 is valid for grains with internal disorder, such thatthe electron motion inside the grains is chaotic. Whenthe electron hops from grain to grain, its momentum isnot conserved and this leads to the finite conductivity 0,Eq. 2.1. The conductance of the contact g depends onthe microscopic properties of the contact, and we con-sider it in most cases as a phenomenological dimension-less parameter controlling the behavior of the system.Note that the model of a periodic array assumes novariation in tunneling conductances.

Upon decreasing the temperature Coulomb interac-tions become relevant and Eq. 2.1 no longest holds.Below we discuss Coulomb effects in more detail andfind that their manifestation in granular systems may dif-fer noticeably from that in homogeneously disorderedmetals.

2. Metallic regime

In the metallic regime electrons tunnel easily fromgranule to granule. The time 0 that the electron spendsinside a grain plays an important role in the metallic

regime: the corresponding characteristic energy =01 is

related to the tunneling conductance and the meanenergy-level spacing as

=g. 2.2

The energy can also be interpreted as the width of thesmearing of the energy levels in the grains. In the limitof large conductances g1 this width exceeds the en-

ergy spacing and the discreteness of the levels within asingle grain ceases to be relevant.Since the electron motion on scales well exceeding the

granule size is always diffusive even in the case of theballistic electron motion inside each grain, the electronmotion on the time scales larger than 1 can be de-scribed by the effective diffusion coefficient Deff relatedto as

Deff= a2. 2.3

Then the Einstein relation gives the conductivity of thegranular sample as

0 = 2e2Deff. 2.4

For periodic arrays Eq. 2.4 is equivalent to Eq. 2.1,which can be seen from Eqs. 1.3, 2.2, and 2.3. At thesame time Eq. 2.4 is more general than Eq. 2.1 sincewith the properly defined diffusion constant Deff it ap-plies to arrays with arbitrary grain arrangement as well.

The energy scale plays a very important role; manyphysical quantities have qualitatively different behaviordepending on whether they are dominated by energieshigher or lower than .

From the experience with homogeneously disorderedmetals one can envision two major causes that may alterthe classical conductivity 0 in Eq. 2.1: i electron-

electron interactions Altshuler and Aronov, 1985; Leeand Ramakrishnan, 1985 and ii quantum interferenceeffects Abrahams et al., 1979; Gorkov et al., 1979. Ac-cordingly, constructing the theory of granular conductorswith reference to the highly advanced theory of disor-dered metals, one can expect two corresponding distinctcorrections to 0. Since in the metallic domain electronseffectively screen out the on-site Coulomb interactions,the bare magnitudes of which are high because of thesmall sizes of the grains, the notion of interaction cor-rections to the conductivity is well justified.

To gain a qualitative understanding of interaction ef-fects, we introduce the characteristic interaction tempo-

ral scale T

/T and the corresponding spatial scaleLTDeff/T associated with this time. One expects thatthe behavior of the interaction correction is different ondistances exceeding the granule size LTa and withinthe granule LTa. Using Eq.2.3 to find the effectivediffusion coefficient Deff, one immediately sees thatthese conditions imply the existence of two distinct tem-perature regions T and T with respect to interac-tion contributions. Accordingly, the correction to theconductivity due to Coulomb interaction can be writtenas a sum of contributions coming from high, , andlow, , energies. This separation of contribution




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naturally follows from the diagrammatic approachwhere the two contributions in question are representedby two distinct sets of diagrams. Deferring the details forlater, the result is as follows: denoting the correctionscoming from the high and low energies with respect to as 1 and 2, respectively, we write

= 0 + 1 + 2 2.5

with1

0=

1

2dgln gEc

maxT, 2.6

Efetov and Tschersich, 2002, 2003 and

2

0=

122gT

, d = 3

1

42gln

T, d = 2

4g

T, d = 1,

2.7

where 1.83 and 3.13 are numerical constantsBeloborodov et al., 2003. The high-energy contribution1 in Eq. 2.5 contains the dimensionality of the arrayd merely as a coefficient and is, in this sense, universal.On the contrary, the low-energy contribution 2 in Eq.2.5 has a different functional form for different arraydimensionalities. Note that for the three-dimensionalcorrection we have kept the temperature-dependentpart only.

At high temperatures T the correction 1, Eq.2.5, grows logarithmically with decreasing tempera-ture. Upon further lowering the temperature this correc-

tion saturates at T and remains constant at T.Then the correction 2 in Eq. 2.7 from the low-energyscales, , where coherent electron motion on scaleslarger than the grain size a dominates the physics, andwhich is similar to that derived for homogeneous disor-dered metals Altshuler and Aronov, 1985, comes intoplay. In the low-temperature regime it is this term thatentirely determines the temperature dependence of theconductivity. At the same time, the contribution 1, al-though temperature independent, still exists in this re-gime and can even be larger in magnitude than 2.Equation 2.6 can be written for T in the form Eq.1.2 which has been observed in a number of experi-

ments Simon et al., 1987; Fujimori et al., 1994; Gerber etal., 1997; Rotkina, 2005 see, e.g., Fig. 5.Rewriting the low-energy contribution 2 in terms of

the effective diffusion coefficient Deff from Eq. 2.3 onereproduces the Altshuler-Aronov corrections Altshulerand Aronov, 1985. This reflects the universal characterof large-scale behavior for a disordered system, and weshow rigorously in Sec. II.D that indeed the granularmetal model can be reduced to an effective disorderedmedium on distances much larger than a single-grainsize i.e., coming from the low-energy, T, excita-tions. The contribution 1, which is dominated by the

energies , is the consequence of and specific to thegranularityand does not exist in homogeneously disor-dered metals.

Now we turn to quantum interference weak-localization effects that exist also in systems withoutany electron-electron interaction. In the metallic regime,where perturbation theory holds with respect to the in-verse tunneling conductance 1/g, the interaction andweak-localization corrections can be considered inde-pendently in leading order.

The weak-localization correction is of a purely quan-tum origin: it stems from the quantum interference of

electrons moving along self-intersecting trajectories andis proportional to the return probability of an electrondiffusing in a disordered medium. In one- or two-dimensional conductors the probability of returning in-finitely many times is unity and the returning trajectoriescan be infinitely long. Thus fully coherent electronpropagation would lead to a divergent weak-localizationcorrection. The finite phase relaxation dephasing timeprevents long trajectories and limits the correction.Adapting here the concept of the effective disorderedmedium we can assume that the results ofAltshuler andAronov 1985 for homogeneously disordered metals ap-ply to granular metals with the proper renormalization

of the diffusion constant. We introduce the effectivedephasing length L=Deff a, where is thedephasing time, which determines the scale for the inter-ference effects. At low temperatures the dephasing time is large and the decoherence length L can exceedthe size a of a single grain, La. In this regime therelevant electron trajectories pass through many gran-ules and quantum interference effects are similar tothose in homogeneously disordered metals and contrib-ute essentially to the conductivity. With increasing tem-perature the decoherence length L decreases and assoon as it drops to the grain size, L a, the trajectories

FIG. 5. Resistance of 3D granular Al-Ge sample as a functionof temperature on a log-log scale, as measured at zero and 100 kOe field open circles. Sample room-temperature re-sistance is 500 . From Gerber et al., 1997.




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contributing to weak localization traverse to only oneneighbor and the weak-localization correction is sup-pressed. Using Eq. 2.3 we write the last condition sepa-rating the domains of relevance and irrelevance ofweak-localization effects as

1. For 1 or

La the quantum interference effects are important,while for

1 or La the corresponding correc-tion drops rapidly with temperature.

The final result for the quantum interference correc-tions reads Beloborodov et al., 2004b; Biagini, Canera,et al., 2005

WL

0=

1

42gln 2.8

for granular films and

WL

0=

1

2g

1/2 2.9

for granular wires. In both equations is the dephasing

time. Within this time the wave function retains its co-herence. The most common mechanism of dephasing,the electron-electron interactions, gives for the dephas-ing time Altshuler et al., 1982

1 =

T

g, d = 2

T2g

1/3 , d = 1. 2.10In this case the condition

1 defines in the 2D caseyet another characteristic energy scale T*=g2, which

marks the interval of relevance of weak-localizationeffects.

The quantum interference correction WL is sup-pressed by applying even a relatively weak magneticfield; the dependence upon the magnetic field can serveas a test for identifying weak-localization effects. At suf-ficient fields, the main temperature dependence of theconductivity will thus come from electron-electron inter-action effects, Eqs. 2.6 and 2.7.

Both electron-electron interactions and quantum in-terference effects decrease the conductivity of a granularsystem at low temperatures, as in homogeneously disor-dered systems. The important feature is that the granu-

larity restrains screening, thus enhancing the role ofCoulomb interaction: this is reflected by the contribu-tion 1 to the conductivity which is specific to granularconductors but absent in homogeneously disorderedmetals. As a result, in 3D and, to some extent, in 2D;see below granular systems the Coulomb interactionscan become the main driving force of a metal-insulatortransition.

This question is worth a more detailed discussion. Thegranular contribution 1 comes from short distancesand is actually the renormalization of the tunneling con-ductance between the grains:

gg =g 1

2dln gEc

maxT, . 2.11

Using renormalization-group methods one can showthat Eq. 2.11 represents the solution of therenormalization-group equation for the effective con-ductance g rather than a merely perturbative correction.As such, it holds not only when the second term on theright-hand side of Eq. 2.11 is much smaller than thefirst one but in a broader temperature regionas long asthe renormalized conductance is large, g1. It is impor-tant that the logarithm in Eq. 2.11 saturates at tem-peratures of the order of . Then one sees from Eq.2.11 that the renormalized conductance g may remainlarge in the limit T0 only provided the original bareconductance g is larger than its critical value

gc = 1/2dlnEc/. 2.12

Ifggc, the effective conductance g renormalizes downto zero at finite temperature. Of course, as soon as gbecomes of the order of unity, Eq. 2.11 is no longer

valid, but it is generally acceptedin the spirit of therenormalization-group approachthat conductanceflow to low values signals electron localization a recentexact solution for a model equivalent to a single grainconnected to a metallic contact lends confidence to thisconclusion Lukyanov and Zamolodchikov, 2004. Theresult 2.12 can also be obtained by analyzing the sta-bility of the insulating state, as discussed below.

The physical meaning of the critical conductance gc ismost transparent for a 3D system: in this case, there areno infrared divergencies and both the localization andAltshuler-Aronov corrections are unimportant even inthe limit T0. This implies that as long as ggc the

granular conductor remains metallic.At ggc the conductance renormalizes to low valuesat temperatures exceeding , thus signaling the develop-ment of the Coulomb blockade. Upon further tempera-ture decrease the system resistance begins to grow expo-nentially at a certain characteristic temperature Tch,indicating the onset of insulating behavior. The tempera-ture Tch tends to zero in the limit ggc. Thus in threedimensions the value gc marks the boundary betweenthe insulating and metallic states at T0.

The interpretation of the critical value gc is lessstraightforward in the case of granular films, since at

ggc the low-temperature conductivity corrections due

to interaction and localization effects diverge logarith-mically. This means apparently that the system becomesan insulator without a sharp transition. Yet the notion of

gc still makes sense as a marker distinguishing betweensystems that are strong Coulomb insulators at low tem-peratures ggc and those that are weak insulatorsggc.

The above results have been obtained within themodel of a periodic granular array cubic lattice, ne-glecting dispersion of the tunneling conductance. In re-ality, typical granular samples are disordered. It is thenimportant to understand what effect the irregularities in




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the granules arrangement may have. It is plausible thatthe universal regime of low energies is hardly affectedby the irregularities since all physical characteristics inthis regime can be expressed through the effective diffu-sion coefficient of the medium. At the same time, thephysical results that are controlled by the local physics,in particular, the critical value gc, are sensitive to thegrain arrangements. In some cases the irregularities canbe incorporated at almost no expense. For example,breaking some finite fraction of junctions between thegranules merely changes the average coordination num-ber z, which appears as the 1/2d factor in Eqs. 2.11and 2.12 for the critical conductance. This means thatthe dispersion in tunneling conductance is not expectedto change noticeably, for example, the position of themetal-insulator transition which may acquire a percola-tion character, and can be taken into account by replac-ing 2d in Eqs. 2.11 and 2.12 with the effective coor-dination number zeffz.

In general, proper treatment of disorder in the grainarrangement may appear more tedious. Yet we do notexpect that it can change the physics of the metallic state

qualitatively. Irregularities of grain displacements and ofother quantities characterizing the system play a muchmore important role in the insulating regime, which webriefly review in the following section.

3. Insulating regime

We begin with a periodic granular array which, in theregime of weak coupling between the grains, is an exem-plary Mott insulator at low temperatures. The electrontransport is mediated by electron hopping from grain tograin. However, leaving a neutral grain and entering itsneighbor involves a considerable electrostatic energycost, and electron transport is blocked at low tempera-tures by the Coulomb gap in the electron excitationspectrum M. At very small tunneling conductances thisgap is simply the Coulomb charging energy of the grain,M= Ec. Virtual electron tunneling to neighboring grainsleads to a reduction of the Mott gap M and, in the limitof noticeable tunneling, the gap decays exponentially in

g Beloborodov, Lopatin, and Vinokur, 2005 until itreaches the inverse escape rate from a single grain . AtM the system falls into the regime of weak Cou-lomb correlations and the insulator-metal transition oc-curs in three dimensions. Using M to estimate thetransition point, one arrives within logarithmic accuracyat the same result for the critical conductance gc as thatderived from renormalization-group considerations seefor details Sec. II.F.

The presence of the hard gap in the excitation spec-trum leads to an activation dependence Arrhenius lawof the conductivity on temperature:

eM/T, TM. 2.13

Indeed, the finite-temperature conductivity is due toelectrons and holes that are present in the system as realexcitations. Their density is given by the Gibbs distribu-

tion, which results in an exponential dependence of theconductivity, Eq. 2.13.

However, the activation behavior is usually not ob-served in real granular samples at low temperatures. In-stead, the experimentally observed resistivity follows thelaw Eq. 1.1, which resembles the Efros-Shklovskii lawderived for doped semiconductors. The fact that the ob-served conductivity behavior cannot be explained interms of the periodic model suggests that disorder playsa crucial role in the low-temperature conductivity in theinsulating state.

The stretched-exponential Shklovskii-Efros-like con-ductivity behavior in granular conductors remained achallenging puzzle for a long time. Several explanations

were advanced; in particular, it was proposed Abeles etal., 1975 that random variations of the capacitance re-sulting from the grain-size dispersion could provide thedependence of Eq. 1.1. However, capacitance disordercan never lift the Coulomb blockade in a single graincompletely and therefore cannot give rise to a finite den-sity of states at the Fermi level Pollak and Adkins,1992; Zhang and Shklovskii, 2004. Furthermore, thestretched-exponential dependence, Eq. 1.1, was ob-served in periodic arrays of quantum dots Yakimov etal., 2003 and artificially manufactured metallic periodicgranular systems Tran et al., 2005 see Fig. 6, wherethe size of the granules and the periodicity in the dot

arrangement were controlled within a few percent accu-racy. Neither of those systems possesses any noticeablecapacitance disorder, yet the dependence 1.1 is ob-served. This indicates that electrostatic disorder unre-lated to grain-size variations but caused most probablyby charged defects in the insulating matrix or substrateis responsible for lifting the Coulomb blockade and forthe finite density of states near the Fermi level resultingin the dependence 1.1.

There is, however, another ingredient necessary forthe variable-range-hopping VRH type of conductivityto occur apart from a finite density of states at energies

FIG. 6. Conductance g0 vs inverse temperature T1/2 for peri-odic granular multilayer and thick film shown in Fig. 2. Inset:for the high-temperature range g0 has been replotted as a func-tion ofT1, indicating Arrhenius behavior from 100 to 160 K.From Tran et al., 2005.




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close to the Fermi level: a finite, although decaying ex-ponentially with distance, probability for tunneling tospatially remoteand not only to adjacentstates closeto the Fermi level. Accordingly, the problem of the hop-ping transport in granular conductors is twofold and amodel should contain i an explanation of the origin ofthe finite density of states near the Fermi level and therole of the Coulomb correlations in forming this densityof states, and ii quantitative description of the mecha-nism of tunneling over long distances through a densearray of metallic grains.

The behavior of the density of states, as mentionedabove, can be caused by an on-site random potential,which in turn is induced by carrier traps in the insulatingmatrix in granular conductors. Traps with energies lowerthan the Fermi level are charged and induce a potentialof the order of e2 /r on the closest granule, where isthe dielectric constant of the insulator and r is the dis-tance from the granule to the trap. This is similar toCoulomb blockade energies due to charged metallicgranules during the transport process. Such a mecha-nism was considered by Zhang and Shklovskii 2004. In

2D granular arrays and/or arrays of quantum dots, onecan expect that the induced random potential originatesalso from imperfections and charged defects in thesubstrate.

We model the electrostatic disorder via the randompotential Vi, where i is the grain index. Such a potentialgives rise to a flat bare density of states at the Fermilevel. In complete analogy with semiconductors, thebare density of states should be suppressed by the long-range Coulomb interaction Efros and Shklovskii, 1975;Shklovskii and Efros, 1988.

Next we consider electron hopping over distances wellexceeding the average granule size in a dense granular

array. This process can be realized as tunneling via vir-tual electron levels in a sequence of grains. Virtual elec-tron tunneling through a single granule or a quantumdot the so-called co-tunneling was first considered byAverin and Nazarov 1990, where two different mecha-nisms for charge transport through a single quantum dotin the Coulomb blockade regime were identified;namely, there are elastic and inelastic co-tunnelingmechanisms. In the course of elastic co-tunneling thecharge is transferred via the tunneling of an electronthrough an intermediate virtual state in the dot such thatthe electron leaves the dot with the same energy as itcame in with. In the latter mechanism inelastic co-

tunneling an electron that comes out of the dot has adifferent energy from that of the incoming one. Afterinelastic co-tunneling the electron leaves behind in thegranule electron-hole excitations absorbing the incom-ing and outgoing energy differences. Note that boththese processes are realized via classically inaccessibleintermediate states, i.e., both mechanisms occur in theform of charge transfer via a virtual state. The inelasticco-tunneling dominates attemperatures larger than T1Ec Averin and Nazarov, 1990.

These two co-tunneling mechanisms can be general-ized to the case of multiple co-tunneling through several

grains. The tunneling probability should fall off expo-nentially with the distance or the number N of granulesleft behind Beloborodov, Lopatin, and Vinokur, 2005;Feigelman and Ioselevich, 2005; Tran et al., 2005, whichis equivalent to the exponentially decaying probabilityof tunneling between states near the Fermi surface inthe theory of Mott, Efros, and Shklovskii Efros andShklovskii, 1975; Shklovskii and Efros, 1988.

Thus hopping processes in amorphous semiconductorsand granular materials are alikeup to the specific ex-pressions for the localization lengthsand by minimiz-ing the hopping probability in the same manner as in theclassic Efros-Shklovskii work, one ends up with the hop-ping conductivity in the form

exp T0/T1/2, 2.14

where T0 is a characteristic temperature depending onthe particular microscopic characteristics. Explicit ex-pressions for this temperature in different regimes aregiven in Sec. II.G. Equation 2.14 explains the experi-mentally observed conductivity behavior in poorly con-ducting granular materials see Eq. 1.1.

Our discussion of variable-range hopping via virtualelectron tunneling through many grains is based on theassumption that the hopping length r* exceeds the sizeof a single grain a. This length decays when the tempera-ture increases and reaches the grain size at some char-

acteristic temperature T. Then the VRH picture nolonger applies; the hops occur between adjacent gran-ules only. Once the probability of a single jump is de-fined, the quantum effects can be neglected and one canuse a classical approach. In particular, at temperatures

TT one expects the conductivity to follow the simpleArrhenius law.

The classical approach to transport in granular metalswas developed by Middleton and Wingreen 1993 andJha and Middleton 2005. One of the results of theirstudy is the presence of a threshold voltage below whichthe conductivity is exactly zero at T=0. The existence ofsuch a threshold voltage is a consequence of the classicalapproach where multiple co-tunneling processes are nottaken into account.

In order to match the results of the classical and hop-ping theories one would have to generalize the approachof Middleton and Wingreen 1993 to include multipleco-tunneling processes. Development of such a theory,in our opinion, is an interesting and important task.

B. Model and main theoretical tools

From the theoretical point of view a granular conduc-tor is an appealing exemplary system whose behavior isgoverned by a nontrivial interplay of electron-electroninteractions, disorder, and quantum fluctuations. Thepowerful approaches that allowed for recent break-throughs in our understanding of the physics of granularmedia are based on effective field theories; in particular,the phase action technique appeared suitable. For situa-tions that fall outside its range of applicability, the ap-




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propriate diagrammatic techniques, generalizing thosefor homogeneously disordered systems, can be devel-oped. In this section we present in detail a model for thedescription of granular metals and introduce bothcomplementary methods, which serve as a foundationfor the quantitative description of granular conductors.

1. Hamiltonian

We model the granular system as an array of metallicparticles connected via tunneling contacts. The Hamil-

tonian H describing a granular conductor has the form

H = i

H0,i + HI+ H t, 2.15

where H0,i stands for the Hamiltonian of noninteracting

electrons in grain i, HI represents the interactions, and

H t describes the electron tunneling between the grains.We now discuss each term in Eq. 2.15.

The H

0,i term describes the free electrons within eachgrain in the presence of impurities,

H0,i = ir 2

2m+ uir irdr , 2.16

where ir and ir are the electron creation and an-

nihilation operators, is the chemical potential, anduir is the disorder potential responsible for the elec-tron scattering inside the ith grain. We adopt the Gauss-ian distribution for uir with pair correlations,

uirujr =

1

2impr rij. 2.17

Throughout, we assume that all grains are in the diffu-sive limit, i.e., the electron mean free path l within eachgrain is smaller than the grain size a. This assumptionsimplifies our calculations because it allows us to avoidconsidering the electron scattering from the grainboundaries, which becomes the main disorder mecha-nism in the case of ballistic grains. At the same time,most of the results obtained in the diffusive limit areexpected to hold also for ballistic grains with an irregu-lar surface as long as the single-grain diffusion coeffi-cient D0 does not enter the final result. This happens in

normal grains provided all relevant energies are smallerthan the Thouless energy ETh of a single grain, Eq. 1.5.Actually, for typical grain sizes of the order 100 themean free path l is comparable with the granule size a.

A single grain may be considered within the standarddiagrammatic approach developed for disordered metalsAbrikosov et al., 1965. The main building block for thediagrams is the single-electron Greens function aver-aged over disorder and it can be derived using the self-consistent Born approximation SCBA. The diagramshown in Fig. 7a represents the relevant contributionto the self-energy:

G0p = i p + isgn2imp 1

. 2.18

Another important block for the diagrammatic tech-nique is the diffusion propagator diffusion which is justthe impurity-averaged particle-hole propagator. In bulkdisordered metals it is given by

D,q =1

D0q2 +

, 2.19

where D0 is the classical diffusion coefficient and isthe Matsubara frequency.In a grain, however, the term q2 has to be replaced by

the Laplace operator with the proper boundary condi-tions Efetov, 1983, 1997; Aleiner et al., 2002. This pro-cedure leads to the quantization of diffusion modes andto the appearance of the lowest excitation energy of theorder of D0 / a2 with a being the grain size. This is justthe Thouless energy ETh that was defined in Eq. 1.5.

The Thouless energy of a nanoscale grain is large andthis allows us to simplify calculations by neglecting allnonzero space harmonics in the diffusion propagatorEq. 2.19. Throughout we assume that ETh is the largest

energy scale associated with our system and use thezero-dimensional approximation for the diffusion propa-gator,

D = 1/ . 2.20

Now we turn to the description of the electron-

electron interaction, the second term HI in the Hamil-tonian Eq. 2.15. We begin by considering interactioneffects in an isolated grain and then turn to a discussionof the intergranular terms which are important becauseof the long-range character of the Coulomb interaction.

The most general form of the electron-electron inter-

action HI in an isolated grain is

HI0 =

1

2 p,q,r,sHpqrsp,

q, r,s,, 2.21

where the subscripts p, q, r, and s stand for states in thegrains, and and label electron spins.

The matrix Hpqrs is a complicated object, but in thelow-energy region of the parameters not all of the ma-trix elements are of the same order. In the case of disor-dered grains that we consider, only the elements withpair indices survive Aleiner et al., 2002; all others aresmall in the parameter 1/g0, where g0 is the single-grain

FIG. 7. Self-energy of the electron Greens function averagedover the impurity potential inside the grains and over tunnel-ing elements between the grains. Averaging over the impuritypotential is represented by a the dotted line while tunneling

elements are represented by b crossed circles.




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13/50

lations vanish. As we will see, the constant gij is nothingbut the tunneling conductance of the contact.

Equation 2.27 is written for time-reversal-invariantsystems orthogonal ensembles. This, in particular,means that there is no magnetic field and/or no magneticimpurities present and therefore all eigenfunctions of aninsulated grain can be chosen to be real. In the limit ofcomparatively strong magnetic fields, one arrives at theunitary ensemble and the first term in the correlationfunction of the matrix elements in Eq. 2.27 vanishes.However, for metallic grains of the order of 510 nm insize the characteristic magnetic field that requires a de-scription in terms of the unitary ensemble is of the orderof several teslas. Thus in the subsequent discussion weuse Eq. 2.27, assuming that the applied magnetic fieldsare not that high.

To conclude this part, we have reformulated the initialtheory with impurities and regular tunneling matrix ele-ments in terms of a model with random tunneling ele-ments. This equivalence holds for zero-dimensionalgrains, or for situations when all characteristic energiesare smaller than the Thouless energy ETh, Eq. 1.5.

Working with random tunneling elements turns out tobe the more convenient approach.

The model introduced in this section, Eqs.2.152.17, 2.23, 2.24, and 2.27, describes astrongly correlated electronic disordered system whichcannot be solved exactly. Below we introduce twocomplementary approaches to explore this problem indifferent regimes. One of these approachesthe dia-grammatic techniqueis especially useful if a perturba-tive expansion is possible with respect to one of the rel-evant parameters. The second method is based on thegauge transformation, which allows us to eliminate theexplicit Coulomb interaction term in the Hamiltonian at

the expense of an appearing phase field. As shown be-low, this approach offers a very powerful tool in its do-main of applicability.

2. Diagrammatic technique for granular metals

There are two routes to construct a diagrammatictechnique for granular metallic systems. First, one canwork in the basis of the exact single-grain eigenfunctionsand use the distribution of the tunneling elements 2.27for the description of scattering between the differentstates of neighboring grains. In this case the impuritiesinside each grain are treated exactly, and by definition

the diagrams that represent impurity scattering withineach grain do not appear in this representation. The al-ternative approach is equivalent to the conventionalcross technique: one begins with the momentum repre-sentation and then carries out the standard averagingover impurities within each grain. Tunneling betweengrains can then be viewed as an intergranular scattering.The corresponding matrix elements, as in the firstmethod, can be viewed as Gaussian random variables.We prefer to follow the latter approach since it isstraightforwardly related to the standard diagrammatictechnique for homogeneously disordered metals, mak-

ing it easier to make comparisons to well-known situa-tions in disordered metals when available.

Following Abrikosov et al. 1965 we construct the ac-

tion expansions with respect to both types of disorder:potential disorder within each grain ur and the inter-granular scattering matrix elements tpq. Shown in Fig. 7are the lowest-order diagrams representing both contri-butions to the self-energy of the single-electron Greensfunction in the granular metals obtained by averagingover ur and tpq. The diagram a describes the potentialscattering within a single grain, while diagram b repre-sents intergranular scattering. Both processes result in asimilar contribution proportional to sgn to the elec-tron self-energy. This shows that on the level of thesingle-electron Greens function the intergranular scat-tering results merely in the renormalization of the relax-

ation time ,

1 = 01 + 2d , 2.28

where 0 is the electron mean free time in a single grain.The next step is to consider the diffusion motion of

electrons through the granular metal. The diffusion mo-tion inside a single grain is given by the usual ladderdiagram that results in the diffusion propagator D,Eq. 2.20. Tunneling between grains does not changethe selection rules for the diagrams. Typical diagramsare shown in Fig. 8b. The diagrams are generated byconnecting the tunneling vertices, and only diagramswithout the intersection are to be kept. This is similar towhat one has in the standard impurity technique Abri-kosov et al., 1965. In order to derive the total diffusionpropagator one should sum up the ladder diagramsshown in Fig. 8b. For the periodic array of the grainswe arrive at the following diffusion propagator D,qBeloborodov et al., 2001:

D,q = 1 + q1 , 2.29

where

FIG. 8. Diagrams represent the Dyson equation a for asingle-grain diffusion propagator Eq. 2.20 and b for thewhole granular system Eq. 2.29. Dotted lines represent theimpurity scattering while crossed circles stand for intergranulartunneling elements.




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q = 2a

1 cos q a, 2.30

a are the superlattice vectors and q is the quasimomen-tum. In Eq. 2.29 only the zero-space harmonics of thediffusion motion in the grain is taken into account. Thisapproximation is valid in the limit

ETh, 2.31

where the energies and ETh are given by Eqs. 2.2 and1.5, respectively. In the limit of small quasimomentaqa1 we have qa2q2 such that the propagator2.29 describes the diffusion motion on scales muchlarger than the size of a single grain a with an effectivediffusion coefficient

Deff= a2 . 2.32

The other necessary blocks of the diagrammatic tech-nique can be constructed analogously to the diffusionpropagator.

3. Coulomb interaction and gauge transformation

Taking an alternative route, one can eliminate the ex-plicit Coulomb term in the model for granular metals,Eqs. 2.152.17, 2.23, 2.24, and 2.27, by introduc-ing a new phase field via a gauge transformation.

To introduce this approach it is convenient to adoptthe formalism of functional integration, which allows

calculations with the Hamiltonian H, Eq. 2.15, to bereplaced by computation of a functional integral overclassical fermion fields iX and their complex conju-gate i

*X, where X= r ,. These fields must satisfy thefermionic antiperiodicity condition

i = i+ , 2.33

where =1/T is the inverse temperature. We define thefield iX such that it is different from zero only in theith grain.

The Lagrangian L entering the functional integralfor the partition function Z,

Z= exp 0

LdD, 2.34takes the form

L = i L0i + Lt + Lc, 2.35where

L0i = i*X

2

2m+ ur iXdr ,

2.36

Lt = i,j;p,q

tij;pqpi* qj, 2.37

and

Lc =e2

2 ijniCij

1nj, 2.38

with ni =i*XiXdr the density field of the grain i.

The Coulomb interaction Lc, Eq. 2.38, can be de-coupled using a Gaussian integration over the auxiliary

field Vi Efetov and Tschersich, 2003,

expe2

2 ij 0

niCij1njd= expi

i

0

ViiXiXdX

ZV1

exp SVdVi , 2.39

where Vi is the bosonic field obeying the boundary

condition Vi = Vi+ and ZV is the partition func-tion,

ZV =

exp SVdVi.

The action SV has the following form:

SV =1

2e2

0

dij

ViCijVj . 2.40

We see from Eqs. 2.322.39 that the Lagrangian be-comes quadratic in fields after the decoupling, Eq.2.39. Instead of dealing with the Coulomb interactionLc, Eq. 2.38, one should consider an effective La-

grangian L0ieff, V for the grain i:

L0i

eff

,V

= L

0i

i

0

Vi

i

id.

2.41

The effective action L0ieff, V is now expressed in terms

of electron motion in granular matter in the presence of

the fluctuating potential Vi of the grains.

We remove the field Vi from the Lagrangian

L0ieff, V, Eq. 2.41, using a gauge transformation of

the fermionic fields

ir, eiiir,, i = Vi, 2.42

where the phases i depend on imaginary time butnot on the coordinates inside the grains. This is a conse-

quence of the Coulomb interaction, Eqs. 2.23 and2.38. Since the action of an isolated grain is gauge in-variant, the phases i enter the Lagrangian of the sys-tem only through the tunneling matrix elements

tij tijeiij, 2.43

where ij =ij is the phase difference of theith and jth grains.

At first glance, the transformation Eq. 2.42, has re-

moved completely the potentials V from the effective

Lagrangian L0ieff,V, Eq. 2.41. In fact, this is not the




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case since the gauge transformation as defined in Eq.2.42 violates the antiperiodicity condition Eq. 2.33,and the resulting effective Lagrangian for the singlegrain changes. In order to preserve the boundary condi-tion Eq. 2.33, certain constraints should be imposed on

the phases i and potentials Vi; namely, the phasesshould obey the condition

i = i+ + 2ki, 2.44

which leads to the following constraint for the potential

Vi:

0

Vid= 2ki , 2.45

where ki = 0 , 1 , 2 , 3 , . . . .The constraint on the phases i, Eq. 2.44, can be

reformulated via introducing a function i assumingarbitrary real values from to and obeying the pe-riodicity condition

= + . 2.46

With the aid of this function the phase i can be writ-ten in the form

i = i + 2Tki, 2.47

which satisfies Eq. 2.44. The potential Vi, in its turn,is taken in the form

Vi = i + Vi, 2.48

where the static variable i varies in the interval

T i T, 2.49

and the dynamic variable Vi satisfies the constraint2.45. Note that the static part of the potential i cannotbe gauged out, contrary to the dynamic contribution

Vi, and the effective Lagrangian assumes the formL0i

eff,. In the limit of not very low temperatures

T , 2.50

the static potential i drops out from the fermionicGreens functions and does not influence the system be-havior. This can be understood by noticing that at mod-erately high temperatures 2.50 the discreteness of thelevels in the grains is important. Then, in the Greens

functions one may replace the variable p , wherep are eigenenergies and is a continuous variable vary-ing from to . Integrating over one may shift thecontour of the integration into the complex plane andremove i provided it obeys the inequality 2.49. Moredetails have been presented by Efetov and Tschersich2003. Note that the variable i cannot be neglected inthe Lagrangian L0i

eff, in the limit of very low tem-peratures.

The integer ki in Eqs. 2.44, 2.45, and 2.47 repre-sents an extra degree of freedom related to the chargequantization and is usually called the winding number.

The physical meaning of the winding numbers becomesespecially clear in the insulating regime where they rep-resent static classical electron charges.

Thus at not very low temperatures, Eq. 2.50, one canwrite the partition function Z in the form

Z= exp L0 + L1, + L2dDD, 2.51

where

L0 = i

L0i , 2.52

with L0i from Eq. 2.36. The tunneling term L1,is

L1, = ij

0

dtij;pqip* jqexpiij ,

2.53

and the term L2 describing the charging effects reads

L2 = ij

Cij

2e2di

d

dj

d. 2.54

One should integrate Eq. 2.51 over the anticommutingvariables with the antiperiodicity condition Eq. 2.33.The integration over includes integrating over thevariable i see Eq. 2.47 and summation over ki.Equations 2.512.54, 2.46, and 2.47 completelyspecify the model that will be studied in subsequent sec-tions.

4. Ambegaokar-Eckern-Schn functional

The partition function Z, Eqs. 2.512.54, can befurther simplified by integration over the fermion fields. This can hardly be done exactly but one can use acumulant expansion in the tunneling term L1,, Eq.2.53. Of course, even in the absence of tunneling therandom potential in L0, Eqs. 2.52 and 2.36, canmake the problem highly nontrivial. Yet, in the limit ofnot very low temperatures, Eq. 2.50, the problem canpossibly be simplified because of neglecting interference

effects and considering the disorder within the SCBA,taking the Greens functions from Eq. 2.18. We per-form the cumulant expansion in the tunneling termS1, up to the lowest nonvanishing second order, in-tegrating over fermionic degrees of freedom and averag-ing over the tunneling matrix elements with the help ofEq. 2.27.

The resulting action, originally derived by Ambe-gaokar et al. 1982 for a system of two weakly coupledsuperconductors, contains only the phases i but nofermionic degrees of freedom. For granular metals thepartition function Z can be written as




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Z= exp SD, S = Sc + St, 2.55where Sc describes the charging energy,

Sc =1

2e2ij 0

dCijdi

d

dj

d, 2.56

and St stands for tunneling between the grains,

St= gij

0

dd sin2ij ij2 .2.57

The function in Eq. 2.57 has the form

= T2Resin2T + i , 2.58

where +0 and one should take the real part in Eq.2.58. Despite the fact that the above functional wasobtained via an expansion in the tunneling term S1,,its validity is not limited to only the insulating regime

g1. The functional can be used in the metallic regime

at temperatures T

, where

is given by Eq. 2.2.Of course, the phase action loses some informationabout the original model, Eqs. 2.342.38, and its ap-plicability in each particular case has to be carefully ana-lyzed. In general, the functional S, Eqs. 2.552.58,does not apply in cases where the coherent diffusive mo-tion of an electron on the scale of many grains is impor-tant. For example, the weak-localization correction can-not be obtained using this approach.

Moreover, one has to be careful when using this ap-proach, even in the weak-coupling regime g1. For ex-ample, the hopping conductivity in the low-temperatureelastic regime is also beyond the accuracy of the phase

action, Eqs. 2.552.58, since it requires considerationof the elastic multiple co-tunneling processes which thephase action misses.

Yet the phase functional approach is extremely pow-erful for many applications. For example, it enables non-perturbative results for the conductivity to be obtainedin the metallic regime at temperatures T.

C. Metallic properties of granular arrays at not very low

temperatures

In this section, we derive the logarithmic correction tothe conductivity Eq. 2.6 in the temperature regime

T. While this correction has the same origin as theAltshuler-Aronov result derived for homogeneously dis-ordered metals Altshuler and Aronov, 1985; Lee andRamakrishnan, 1985, its form is specific to granularmetals. In contrast to Eq. 2.6, Altshuler-Aronov cor-rections are sensitive to the sample dimensionality andexhibit the logarithmic behavior only in two dimensions.

The logarithmic correction Eq. 2.6, as shown below,can be obtained as a result of the renormalization of thetunneling coupling g due to Coulomb correlations. Itcomes from the short distances, and this explains its in-sensitivity to the sample dimensionality. At tempera-

tures smaller than , coherent electron motion on thescale of many grains becomes important, and the con-ductivity correction acquires a form similar to that forhomogeneously disordered metals.

For the sake of simplicity we consider a periodic sys-tem, assuming that the granular array forms a cubic lat-tice. This implies that all grain sizes and intergranularconductances are equal to each other. At the end of thissection we discuss the applicability of the result 2.6 tosystems containing irregularities. We start with perturba-tion theory in 1/g and then generalize our derivation of

Eq. 2.6 using the renormalization-group approach.

1. Perturbation theory

In the limit of large tunneling conductances g1 thetunneling term St, Eq. 2.57, suppresses large fluctua-tions of the phase . In this regime the tunneling termSt can be expanded in phases ij because they aresmall. Charge quantization effects in this regime are notpronounced, allowing neglect of all nonzero windingnumbers ki. At the same time, phase fluctuations canchange the classical result considerably, Eq. 2.1.

The part of the action S quadratic in i in Eqs.2.552.57 will serve as the bare action for the pertur-bation theory. Keeping second-order terms in i inEqs. 2.552.57 and performing the Fourier transfor-mation in both the coordinates of the grains and theimaginary time we reduce the action S in Eq. 2.55 tothe form

S0 = Tq,n

q,nGq,n1 q,n, 2.59

where

Gqn1 = n

2/4Eq + 2gnq 2.60

and q is given by Eq. 2.30. Here q are the quasi-momenta of a periodic array; Eq is the Fourier trans-form of the charging energy related to the Fourier trans-form of the capacitance matrix Cq as Eq= e2 /2Cq.The conductivity of the grain arrays can be ob-tained from the current-current correlation function viathe standard Kubo formula. The conductivity to leadingorder in 1/g is shown in Fig. 9. Its analytical expressionis given by Efetov and Tschersich, 2003

FIG. 9. The diagram representing the conductivity to leadingorder in 1/g. The crossed circles represent the tunneling ma-trix elements t

kk

ijeiij where phase factors appear from the

gauge transformation.




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=2e2gT2ia2d

0

d1 ei

sin2Texp Ga ,

2.61

where the analytic continuation to the real frequency is

assumed as i, and the function Ga is

Ga = 4Tad

n0 ddq

2d

Gqn sin2 q a

2sin2

n

2,

2.62

where d is the dimensionality of the array. The factor

expGa in Eq. 2.61 is responsible for the interac-tion effects and appears as a result of averaging thephase exponents ei emerging after the gauge transfor-mation Eq. 2.42. One can see from Eqs. 2.60 and

2.62 that the function Ga behaves as a logarithmlngEc at values ofof the order 1/ Tthat are essentialfor the calculation. With logarithmic accuracy one canneglect the n

2 term in Gq,n1 , Eq. 2.60, and reduce Eq.

2.62 to

Ga =T

dgn0

c 1 cosn

n. 2.63

In Eq. 2.63 one should sum over positive Matsubarafrequencies up to the cutoff c gEc. For larger fre-quencies the first term in Eq. 2.60 is no longer smalland there is no logarithmic contribution from these largefrequencies. Equation 2.63 shows a lack of dependenceof the result on the structure of the lattice. It is alsoimportant that there are no infrared divergencies inthe integral over q in any dimensionality including twoand one dimensions.

With logarithmic accuracy, one can replace by 1/Tinthe function Ga and calculate the remaining integralover in Eq. 2.61, ignoring the dependence of the

function Ga on . Taking the limit 0 we obtain theresult 2.6 in the temperature interval T:

= 01 12dg lngEc

T . 2.64

It was shown by Efetov and Tschersich 2003 that termsof the order 1/g2ln2gEc /T are canceled out in theexpansion of the conductivity correction, which meansthat the accuracy of Eq. 2.64 exceeds the accuracy of

the first-order correction. Furthermore, this cancellationis not accidental and the result 2.64 turns out to beapplicable even at temperatures at which the conductiv-ity correction becomes of the same order as 0 itself.This fact is explained in the next section where the tem-perature dependence of the conductivity is consideredwithin the renormalization-group RG scheme.

2. Renormalization group

In order to sum all logarithmic corrections to the con-ductivity we use the RG scheme suggested for the one-

dimensional XY model long ago Kosterlitz, 1976 andused later in a number of works Guinea and Schn,1986; Bulgadaev, 1987a, 1987b, 2006; Falci et al., 1995.As the starting functional we take the tunneling actionSt,

St= gi,j

0

0

dd sin2ij ij2

.

2.65The charging part Sc is not important for the renormal-ization group because it determines only the upper cut-off of integrations over frequencies. In the limit T0the function is proportional to 2 and theaction is dimensionless. We neglect in this section thenonzero winding numbers ki and replace the phasesi by the variables i Eq. 2.47.

Following standard RG arguments we want to deter-mine the change in the form of the action St when thecutoff is changed. Generally speaking, it is not guaran-teed that after integrating over the phases in an inter-val of the frequencies one comes to the same functionsin2 in the action. The form of the functional maychange, which would lead to a functional renormaliza-tion group.

In the present case appearance of terms sin2 2,sin2 4, etc., is not excluded and they are generated inmany-loop approximations of the RG. Fortunately, theone-loop approximation is simpler and the renormaliza-tion in this order results in a change of the effectivecoupling constant g only.

To derive the RG equation we represent the phase in the form

ij= ij0 + ij, 2.66

where the function ij0 is the slow variable and is not

equal to zero in the frequency interval 0c, whilethe function ij is finite in the interval cc,where is in the interval 01. Substituting Eq.2.66 into Eq. 2.65, we expand the action St up toterms quadratic in ij. Integrating Eq. 2.65 for thepartition function

Z= exp StDover the fast variable ij with logarithmic accuracy we

come to a new renormalized effective action St,

St= 2gi,j

0

0

dd

sin2ij ij2

1 2gd

, 2.67where =ln . It follows from Eq. 2.67 that the formof the action is reproduced for any dimensionality d ofthe grain lattice. This allows us to write the followingrenormalization-group equation:




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g

=

1

2d. 2.68

The solution of Eq. 2.68 is simple. Neglecting the Cou-lomb interaction in the action St, Eq. 2.65, is justifiedonly for energies smaller than gEc and this gives theupper cutoff. Then the renormalized conductance gTtakes the form

gT =g 12d

lngEcT 2.69and using Eq. 2.1 we come to Eq. 2.64 for the con-ductivity. Both quantities depend on the temperaturelogarithmically.

Equation 2.69 is obtained in the one-loop approxi-mation and should be valid so long as the effective con-ductance gT remains much larger than unity. This as-sumes that the perturbation theory result, Eq. 2.64, canbe extended to a wider temperature interval and, in fact,it is valid as long as

g 2d1 lngEc/T 1. 2.70

At high temperatures, when the inequality 2.70 is sat-isfied, one can speak of metallic behavior of the system.At lower temperatures TTc, where

Tc =gEcexp 2gd, 2.71

the system is expected to show insulating properties.The result for the conductivity correction 2.64 was

obtained using a cubic lattice model. It is important tounderstand how it will change in the more realistic caseof an irregular array. First, we note that Eq. 2.64 can begeneralized to the case of an arbitrary periodic latticewith the result

= 01 1zg

lngEcT , 2.72where z is the coordination number of the arbitrary pe-riodic lattice.

Dispersion of the tunneling conductance was studiedby Feigelman et al. 2004 within a 2D model that as-sumed regular periodic positions and equal sizes ofgrains but random tunneling conductances. It was shownthat the dependence 2.72 holds in a wide temperaturerange in the case of a moderately strong conductancedispersion. However, the distribution of conductancesbroadens and this effect becomes important close to the

metal-insulator transition; it was suggested to describethe transition in terms of percolation Feigelman et al.,2004. One may expect that in 3D samples the conduc-tance dispersion effect on macroscopic transport will beless important than in 2D ones, while on the contrary itwill be more dramatic in 1D samples.

Finally, we comment on the validity of Eq. 2.72 foran arbitrary irregular array. In the RG approach capaci-tance disorder leads to a local renormalization of tunnel-ing conductances. The general irregular system can beviewed as an irregular array of equal-size grains withrandom tunneling conductances. The main difference

between such a system from that considered byFeigelman et al. 2004 is that the coordination numbervaries from grain to grain. In this case one expects thatthe coefficient in front of the second term in Eq. 2.72will be determined by an effective coordination numberin the system, which is expected to be close to the aver-age number of neighbors of a grain.

Following the RG scheme we did not take into ac-count nonzero winding numbers ki, Eqs. 2.442.47.This is a natural approximation because the contributionof such configurations should be exponentially small in g

expcg, where c is of the order of unity. So long asthe effective conductance g is large in the process of therenormalization, the contribution of the nonzero wind-ing numbers ki can be neglected. They become impor-tant when g becomes of the order of unity. This is theregion of the transition into the insulating state and ap-parently the nonzero winding numbers play a crucialrole in forming this state. However, recent publicationsAltland et al., 2004, 2006; Meyer et al., 2004 have ar-gued that the nonzero winding numbers might becomeimportant in 1D samples at temperatures T* parametri-cally exceeding Tc, Eq. 2.71.

D. Metallic properties of granular arrays at low temperatures

The phase functional technique may not be used forobtaining the conductivity corrections at temperaturesT since in this regime the coherent electron motionover a large number of grains, which is missed in thefunctional approach Ambegaokar et al., 1982, becomesimportant. In this section we derive the conductivity cor-rection using the diagrammatic perturbation theory de-scribed in Sec. II.C.1. Unlike the phase functional ap-proach, this technique does not allow us to obtainnonperturbative results in an easy way but has an advan-tage of being applicable at arbitrary temperatures. In

particular, we show below that it reproduces at Tthe perturbative result Eq. 2.64. The conductivity cor-rection agrees in the low-temperature regime with theone obtained by Altshuler and Aronov 1985, Lee andRamakrishnan 1985, and Belitz and Kirkpatrick 1994for homogeneously disordered samples.

In Sec. II.C.1, we described the main building blocksof the diagrammatic technique including, in particular,the diffusion propagator defined by the ladder diagramin Fig. 8. The same ladder diagram describes the dress-ing of the interaction vertex as shown in Fig. 10a. Thedressed vertex can be used to obtain the polarization

FIG. 10. Diagrams representing a the vertex correction andb the renormalized Coulomb interaction.




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operator that defines the effective dynamically screenedCoulomb interaction Fig. 10b:

V,q = Cqe2

+2q

+ q1 . 2.73

The conductivity of granular metals is given by the ana-lytical continuation of the Matsubara current-currentcorrelation function. In the absence of the electron-electron interaction the conductivity is represented bydiagram a in Fig. 11 which results in the high-temperature Drude conductivity 0 defined in Eq.2.1. First-order interaction corrections to the conduc-tivity are given by diagrams be in Fig. 11. Thesediagrams are analogous to those considered by Altshulerand Aronov 1985 for the correction to the conductivityof homogeneous metals.

We consider the contributions from diagrams b, c

and d, e separately. The sum of diagrams b, cresults in the following correction to the conductivity:

1

0=

1

2dgIm

q dqV,q, 2.74

where = d / dcoth/ 2T, and the potential

V,q is the analytic continuation of the screened Cou-lomb potential V ,q with dressed interaction verticesincluding those attached at both ends,

V,q =2Ecq

q

i4qEcq i

. 2.75

The expression for the screened Coulomb interaction

V,q, Eq. 2.75, is written in a simplified form usingthe fact that the charging energy Ecq = e2 / 2Cq, ex-pressed in terms of the Fourier transform of the capaci-tance matrix Cq, is much larger than the escape rate .

Performing the integration over the frequency andsumming over the quasimomentum q in Eq. 2.74 withlogarithmic accuracy we obtain the correction to theconductivity, Eq. 2.6. One can see from Eq. 2.74 thatthe contribution 1 in Eq. 2.6 comes from the large

energy scales, . At low temperatures T, thelogarithm is cut off by the energy and is no longertemperature dependent.

To obtain the total correction to the conductivity of agranular sample the two other diagrams, d and e inFig. 11, should be taken into account. These diagramsresult in the following contribution to the conductivityBeloborodov et al., 2003:

2

0=

2g

dq d Im

V,qa sin2q a

q i.

2.76

In contrast to the contribution from 1, Eq. 2.74, themain contribution to the sum over the quasimomentumq in Eq. 2.76 comes from low momenta q1/a. In thisregime the capacitance matrix Cq in Eqs. 2.75 and2.76 has the asymptotic form

C1q =2

ad

ln1/qa, d = 1

/q , d = 2

2/q2, d = 3.

2.77

Using Eqs. 2.752.77, we obtain the result for the cor-rection 2 in Eq. 2.7. This correction has a physicalmeaning similar to that of the Altshuler-Aronov correc-tion Altshuler and Aronov, 1985 derived for homoge-neously disordered metals.

Comparing the results in Eqs. 2.52.7 with thoseobtained in the previous section using the Ambegaokar-Eckern-Schn functional one can see that the correctionto the conductivity obtained in Sec. II.C is equivalent tothe correction 1 in Eq. 2.5, which corresponds in thediagrammatic approach to the sum of diagrams b and

c in Fig. 11. The correction 2 in Eq. 2.5 becomesimportant only at low temperatures T where theAmbegaokar-Eckern-Schn functional is not applicable.

Using the final results of the calculations, Eqs. 2.6and 2.7, we now make an important statement aboutthe existence of a metal-insulator transition in three di-mensions. It follows from Eq. 2.7 that for a 3D granu-lar array there are no essential corrections to the con-ductivity at low temperatures T coming from lowenergies , since the correction 2 is always small.This means that the result for the renormalized conduc-tance, Eq. 2.69, for 3D samples can be written withinlogarithmic accuracy in the form

gT =g 1

6ln gEC

maxg,T , 2.78

such that it is valid for all temperatures as long as therenormalized conductance is large, gT1.

One can see from Eq. 2.78 that for a large bare con-ductance, g 1/6lngEC/, the renormalized con-ductance g is always large and the system remains me-tallic down to zero temperatures. In the opposite limit

g 1/6lngEC/, on decreasing temperature the sys-tem flows to the strong-coupling regime, g1, which in-

FIG. 11. Diagrams describing the conductivity of granularmetals: diagram a corresponds to 0 in Eq. 2.5 and it is theanalog of the Drude conductivity. Diagrams be describe

first-order corrections to the conductivity of granular metalsdue to the electron-electron interaction. The solid lines denotethe propagators of electrons and dashed lines describe effec-tive screened electron-electron propagators. The sum of thediagrams b and c results in the conductivity correction 1in Eq. 2.5. The other two diagrams, d and e, result in thecorrection 2. From Beloborodov et al., 2003.




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dicates the onset of the insulating phase. One can seethat with logarithmic accuracy the critical value of theconductance gc is given by Eq. 2.12, and this valueseparates the metallic and insulating states Beloboro-dov et al., 2003.

One can see from Eq. 2.7 that in 1D and 2D samplesthe correction to the conductivity, being negative, growswith decreasing temperature and diverges in the limitT0. This behavior is usually attributed to localizationin the limit T0.

However, the actual situation is more interesting. Re-cently Basko et al., 2005 it was demonstrated that theremust be a metal-insulator transition at finite tempera-ture Tk in systems with weak repulsion, provided allone-particle states are localized. This means that theconductivity is strictly zero at TTk and becomes finiteat TTk. The 1D and 2D granular systems with Cou-lomb interaction discussed here should belong to theclass of models considered by Basko et al. 2005 allone-particle states should be localized in 1D and 2D sys-tems for any disorder and one can expect such a tran-sition. Clearly, the results of Eq. 2.7 should hold forTTk.

At the same time, the 3D samples do not fall into theclass of models studied by Basko et al. 2005 becausethere exist both localized and extended one-particlestates. Therefore for 3D granular systems one can speakof the metal-insulator transition at T=0 only. The tran-sition point can be varied by changing, e.g., the tunnel-ing conductance between the grains or their size.

Finally, we note that the low-temperature conductivitycorrection 2 is le

review - granular electronic systems

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