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tivcreases exponentially with N. Our theory provides a qualitative explanation for the transport properties of

PHYSICAL REVIEW B 69, 035413 ~2004!organic charge-density wave compounds of TCNQ family.

DOI: 10.1103/PhysRevB.69.035413 PACS number~s!: 73.50.Bk, 72.20.Ee, 71.45.Lr, 72.80.Le

I. INTRODUCTION

In recent years electron transport in quasi-one-dimensional ~quasi-1D! systems moved into focus of bothfundamental and applied research. Quantum wires, nanotuberopes, conducting molecules, etc., are being examined aspossible elements of miniature electronics devices. In paral-lel, discovery of quasi-1D structures termed stripes in cor-related electron systems ~high-Tc cuprates, quantum Hall de-vices, etc.!, invigorates efforts to understand unconventionalphases in two and three dimensions starting from models ofweakly coupled 1D chains.

Experimentally, the low-temperature conductivity s(T)of quasi-1D systems is often of the insulating type. Its tem-perature dependence gives information about the nature ofcharge excitations. For example, the activated dependence2ln s(T)}1/T indicates a gap in the spectrum. In quasi-1Dsystems such a gap commonly arises from the Mott-Peierlsmechanism,1,2 where the commensurability with the host lat-tice is crucial. Yet there are many 1D and quasi-1D systemswhere commensurability plays a negligible role. In this situ-ation the jellium model ~an electron gas on a positive com-pensating background! is a good approximation. This is thekind of systems we study in this paper. We will show thattheir low-temperature transport is dominated by a variable-range hopping ~VRH!, which leads to a slower than expo-nential T dependence3,4 of the conductivity. Our theory may

apply to a number of systems, both naturally occurring andman-made. Prototypical 1D examples are individual quan-tum wires on carbon nanotubes. Stripe phases,5,6 quantumwire arrays in heterojunctions,7 carbon nanotube films,8 andatomic wires on silicon surface,9 are two-dimensional ~2D!examples. In three dimensions ~3D!, an important and wellstudied class of quasi-1D compounds is charge-densitywaves1012 ~CDW!.

To characterize the strength of Coulomb correlations in aquasi-1D system we define the dimensionless parameter rs[a/2aB , where a is the average distance between electronsalong the chain direction and aB5\2k/me2 is the effectiveBohr radius. The latter is expressed in terms of the dielectricconstant of the medium k and the electron band mass m. Inpractically all known realizations of 1D and quasi-1D sys-tems, rs exceeds unity, often by orders of magnitude. Belowwe assume that rs@1. Under this condition the dynamics ofelectrons can be treated semiclassically. Neglecting quantumfluctuations altogether for a moment, we arrive at the pictureof electrons forming a classical 1D Wigner crystal in the caseof a single chain @Fig. 1~a!# or an array of such crystals in 2Dand 3D @Figs. 1~b! and ~c!#, with the period along the chainequal to a. Formation of the crystal enables the electrons tominimize the energy of their mutual Coulomb repulsion. Inorder to correctly assess the role of quantum fluctuations~zero-point motion! in such crystals, one has to take intoaccount two circumstances. First is the unavoidable presenceVariable-range hopping in quasi-

M. M.Department of Physics, University of Californ

S. Teber and BWilliam I. Fine Theoretical Physics Institute, Univers

~Received 12 July 2003; revised manuscript rece

We study the effect of impurities on the ground sone-dimensional chain and quasi-one-dimensional syinteractions impose a short-range periodicity of the elecrystal ~or equivalently, a 4kF charge-density wave! icenters. We show that a three-dimensional array of chconcentrations N. At large N, impurities divide the chafrom such rods correspond to charge excitations whoslow temperatures the conductivity is due to the variablthe Efros-Shklovskii ~ES! law, 2ln s;(TES /T)1/2. TESgrowth of s . When N is small, the metallic-rod ~alsostate survives only in the form of rare clusters of atycharge excitations. In the bulk of the crystal the chargecollectively. A strongly anisotropic screening of the Cenergy Coulomb gap and an unusual law of the variparameter T1 remains constant over a finite range ofreplaced by the Mott law, 2ln s;(TM /T)1/4. In the Mdown. Thus, the overall dependence of s on N is nonmpicture applies at all N. The low-temperature conduc0163-1829/2004/69~3!/035413~18!/$22.50 69 0354ne-dimensional electron crystals

oglerSan Diego, La Jolla, California 92093, USA

I. Shklovskiiof Minnesota, Minneapolis, Minnesota 55455, USA

d 27 October 2003; published 29 January 2004!

te and the low-temperature Ohmic dc transport in ams of many parallel chains. We assume that strongon positions. The long-range order of such an electrondestroyed by impurities, which act as strong pinningns behaves differently at large and at small impuritys into metallic rods. Additions or removal of electronsensity of states exhibits a quadratic Coulomb gap. Atrange hopping of electrons between the rods. It obeysecreases as N decreases, which leads to an exponentialnown as interrupted-strand! picture of the groundically short rods. They are the source of low-energycitations are gapped and the electron crystal is pinnedlomb potential produces an unconventional linear in

le-range hopping conductivity 2ln s;(T1 /T)2/5. Thempurity concentrations. At smaller N the 2/5 law istt regime the conductivity gets suppressed as N goesnotonic. In the case of a single chain, the metallic-rodity obeys the ES law, with log corrections, and de-2004 The American Physical Society13-1

M. M. FOGLER, S. TEBER, AND B. I. SHKLOVSKII PHYSICAL REVIEW B 69, 035413 ~2004!of random impurities that act as pinning centers. Second is afinite interchain coupling ~in 2D and 3D systems!. Either oneis sufficient to make the quantum fluctuations of electronpositions bounded. For example, in the case of a singlechain, the slow growth of the zero-point motion amplitudewith distance13,14 is terminated at the nearest strong pinningcenter. In higher dimensions, the zero-point motion ampli-tude is finite even without impurities because of the inter-chain interaction. In fact, renormalization-groupapproaches15 indicate that the Wigner crystal or, equiva-lently, 4kF CDW is the true ground state of a system ofweakly coupled chains starting already from rather modestrs . In all situations, the net effect of quantum fluctuations isto slightly renormalize the bare impurity strength and/or bareinterchain coupling. The calculation of renormalized param-eters is possible via the standard bosonization technique.16For the treatment of, e.g., 1D case, one can consult Refs.13,14,17 and 18. Below such a renormalization is assumed tobe taken into account and it is not discussed further. Hence-forth we will often refer to the systems we study as electroncrystals.

Due to impurity pinning, at zero temperature and in thelimit of small electric field ~Ohmic regime! the conductivityof the electron crystal vanishes. This behavior is common forall pinned systems. It motivated a large body of work19,12devoted to mechanisms of nonlinear transport that become

FIG. 1. Pinned 1D and quasi-1D systems on a uniformlycharged positive background. Dots and tick marks label the posi-tions of electrons and impurities, respectively. ~a! 1D crystal. The1 and 2 signs denote the charge of the metallic rods, which havean average length of l. ~b! An array of decoupled chains in the casel,ls . ~c! An array of coupled chains for the case l.ls . The pre-ferred arrangement of the electrons on neighboring chains may de-pend on the exact geometry of the system. In order not to compli-cate the drawing, we adopted the convention where the ground statecorresponds to the same horizontal positions of electrons on eachchain. The interchain interactions try to diminish the deviationsfrom this ground state leading to dipolar distortions of a character-istic length ls around impurities.03541possible in the presence of a finite electric field, in particular,creep and sliding. In the present context, such mechanismswould involve a collective motion of large numbers of elec-trons that overcome pinning barriers either by thermal acti-vation or by quantum tunneling. It has been understood,however, on some qualitative level,19 that if compact chargeexcitations are allowed by the topology of the system, thensuch excitations would dominate the response at low tem-peratures and would give rise to a nonzero Ohmic conduc-tivity at T.0. Below we will demonstrate that this is indeedthe case in the electron crystals. We clarify the nature of thecompact low-energy charge excitations and propose a theoryof their low-temperature Ohmic transport that consistentlyaddresses the role of long-range Coulomb interactions.20

In this paper we are focused exclusively on the chargetransport and ignore any effects related to the spin degree offreedom. This is legitimate for rs@1 because electron aretightly localized at the sites of the classical Wigner crystaland the energy of their spin-dependent exchange interactionis exponentially small.

We will assume that impurities that pin the crystal arestrong enough to enforce preferred order of electrons nearbyor, in the CDW terminology, the preferred phase. The rela-tion between the phase f and the elastic displacement of thecrystal u is f52(2p/a)u . In Fig. 1 impurities are shownby vertical tick marks and it is assumed that they interactwith nearby electrons by a repulsive potential comparable inmagnitude to the Coulomb interaction energy e2/ka betweennearest electrons on the chain. This condition is sufficient toensure that the impurity to act as a strong pinning center. Anexample of such an impurity is an acceptor residing on thechain. In the ground state one electron is bound to the accep-tor and the electron-acceptor complex ~of total charge e) isbuilt into the crystal, i.e., it is positioned squarely in betweenthe two closest other electrons. One can say that the crystalcontains a plastic deformationa vacancy bound to thenegatively charged acceptor. Overall, the region around theimpurity is electrically neutral.

In the case of a single chain @Fig. 1~a!#, strong impuritiesdivide the crystal into segments, which behave as individualmetallic rods. A charge can easily spread over the length ofeach rod, while it has to tunnel through an impurity to moveto a neighboring rod. Each rod contains an integer number ofelectrons but the charge of the positive background is ran-dom because of the assumed incommensurability. In theground state the distribution of the rods total charges, elec-trons plus background, is uniform between 2e/2 and 1e/2(f is between 2p and p). Larger charges cost more Cou-lomb energy and correspond to charge excitations above theground state. Transitions between ground and excited statesoccur by discrete changes in the number of electrons on therods.

Let be the minimal by absolute value change in theself-energy of a given rod due to change of its charge by oneunit. By the self-energy we mean the Coulomb energy ofinteraction among the electrons on the given rod, charges onall other rods held fixed. We adopt the convention that thechemical potential ~Fermi energy! corresponds to the zeroenergy. In this case, is non-negative ~nonpositive! for ad-3-2

VARIABLE-RANGE HOPPING IN QUASI-ONE- . . . PHYSICAL REVIEW B 69, 035413 ~2004!dition ~subtraction! of the electron. We denote by gB() thedistribution function of averaged over impurity positions.We refer to gB as the bare density of states of charge excita-tions. In Sec. III we show that random distribution of chargesof the rods creates a finite gB(0). Small come from rodswith net charges close to 6e/2. These rods make possible aVRH conductivity at low temperatures.

Consider now a 3D system of parallel chains. Impuritieswith concentration N divide the chains into segments of av-erage length l51/Na

2, where a

2 is the area per chain iny-z plane ~we assume the chains to be along the x direction!.We get two cases distinguished by the relative importance ofinterchain interactions. In the first case @Fig. 1~b!#, the chainsare far enough from each other and/or the impurity concen-tration is large enough so that the interchain coupling overthe length ;l of a typical segment can be neglected in com-parison with its longitudinal compression energy. As a result,the phases of different segments are uncorrelated and thesystem again behaves as a collection of metallic rods. Polar-izability of the rods generates a strongly anisotropic dielec-tric constant. Like for a single chain, in the ground state ofthe system, a finite bare density of states gB at zero energyoriginates from random background charges of the rods.Again this leads to the VRH at low temperatures.

In the other case @Fig. 1~c!#, the concentration of impuri-ties is small and chains are strongly interlocked. The elasticdistortions are concentrated in small regions around indi-vidual impurities ~see below!. Away from impurities thecrystal possesses a good 3D order. The true long-range orderis however absent because of the cumulative effect of smallelastic distortions in a large volume. The elastic displacementfield u (r) of the electron crystal lattice away from impuritiesgradually varies in space. The length scale where its varia-tion is of the order of a ~variation in f is of the order ofunity! is referred to as Larkin length. The Larkin length isexponentially large,21,22 effectively infinite, because the Cou-lomb interactions make the crystal very rigid. This will bediscussed in more detail in Sec. V.

The region near a typical impurity has the followingstructure.23 On one side of the impurity the chain is com-pressed, which creates an excess negative charge; on the op-posite side, it is stretched resulting in a positive charge of thesame absolute value

M. M. FOGLER, S. TEBER, AND B. I. SHKLOVSKII PHYSICAL REVIEW B 69, 035413 ~2004!A. Single chain

In this case, studied in Sec. III, the Coulomb interaction isnot screened. However, in 1D the 1/x decay of the Coulombpotential is on the borderline between the short- and thelong-range interactions. Consequently, most physical quanti-ties differ from their counterparts for the short-range~screened! interaction only by some logarithmic factors. Forexample, the density of states of charge excitations exhibits alogarithmic suppression,30

g~!5gB

ln~e2/kluu!. ~4!

In a first approximation, such a suppression can be disre-garded in the calculation of the VRH transport, namely, onecan assume that g()5gB5const. In this approximation onearrives at the conventional Mott VRH,3,28 which in 1D coin-cides with the ES law of Eq. ~2!. Let us denote by TES

(0) thevalue of TES that one obtains neglecting the Coulomb gap,then TES

(0);1/gBjx . Here jx stands for the localization lengththat determines the asymptotic decay P}exp(22x/jx) of theprobability of tunneling of charge-e excitations over a largedistance x. If the probability of tunneling between nearestrods is written in the form exp(22s), where s@1, then tun-neling paths with returns can be neglected. In this case jx5l/s . Using the expression for s from Ref. 13, we obtain

jx;l

rs1/2ln3/2~ l/a !

, ~5!

TES(0)5C1

e2

kl rs1/2ln5/2~ l/a !. ~6!

In the last equation we absorbed numerical factors into thecoefficient C1;1.

A similar expression for TES(0) was obtained by Nattermann

et al.31 for the model of a disordered Luttinger liquid withshort-range interactions and weak pinning. Our Eqs. ~5! and~6! differ from the corresponding results of Ref. 31 by twologarithmic factors. One of them originates from logarithmiccharging energy of metallic rod; the other, from the loga-rithm in the tunneling action s, see also Refs. 13, 18, and 32.

Once the logarithmic Coulomb gap is taken into account,the T dependence of the conductivity can still be written inthe form of an ES law @Eq. ~2!# but TES becomes a functionof T, as follows:

TES5TES(0)lnS e2

klTES(0)ATES

(0)

T D . ~7!Note that in the 1D case, the standard derivation3 of the VRHlaw ~2! overlooks the role of very resistive hops in someexponentially rare places along the chain. A more carefulapproach shows33,32 that Eq. ~2! and its generalizationthrough Eq. ~7! are valid only if the chain is sufficientlyshort. The quantitative criterion on the length of the chain03541can be obtained following the excellent discussion in Refs.33 and 32. This, however, goes beyond the scope of thepresent work.

B. 3D systems with large impurity concentrations

This case, formally defined by the inequality a!l!ls isstudied in Sec. IV. It may be realized in strongly anisotropicCDW compounds such as KCP where the soliton length ls islarge (102a or so! and/or in samples where a relatively highimpurity concentration is created intentionally34,35 so that l issmall. Possible non-CDW realizations include arrays of rela-tively distant 1D conductors, e.g., quantum wires,nanotubes,36 or polymers.37

As elaborated in Sec. I, impurities divide the system intoa collection of metallic rods. The finite 3D concentration ofhighly polarizable rods results in a large dielectric constant38along the x axis. The Coulomb interaction is thereforestrongly anisotropic but the Coulomb gap remains parabolic,g()}2, as in isotropic systems. Tunneling is anisotropic aswell. The interchain tunneling is accomplished by single-electron-like excitations, which do not perturb charges on theintermediate chains along the tunneling path. In Appendix Bwe estimate the corresponding transverse localization lengthj to be

j5a

ln~e2/kat!, ~8!

where t is the interchain hopping matrix element in thetight-binding band-structure model. For the low-T VRH con-ductivity we again obtain the ES law with TES given by

TES5C2e2

kl F a2 Arsj2 lnS laD G1/3

. ~9!

Here C2 is another numerical factor of the order of unity. Wesee that in both Secs. II A and II B, the ES law looses itsuniversality, because TES depends on the impurity concentra-tion N through l51/Na

2. For a single chain ~Sec. II A! this

dependence originates mainly from the dependence of thelocalization length jx on l. In 3D ~case B!, the full effectivedielectric constant and therefore, the density of states insidethe Coulomb gap depend on N as well. In both cases, withdecreasing N the temperature TES decreases, which at a fixedtemperature leads to an exponentially increasing conductiv-ity.

In doped semiconductors similar violations of the univer-sality of Eq. ~2! are known to occur near the metal-insulatortransition. In that case, however, TES has an opposite depen-dence on N. In particular, TES vanishes when N grows andreaches the critical concentration.4 Similarly, previous theo-ries of the VRH transport in strongly anisotropic systemsdealt with gapped, semiconductorlike materials ~commensu-rate CDW! where impurities provided carriers,39,40 so thatthe conductivity was found to grow with the impurity con-centration. In contrast, our work is devoted to systems,3-4

VARIABLE-RANGE HOPPING IN QUASI-ONE- . . . PHYSICAL REVIEW B 69, 035413 ~2004!which are metallic ~sliding! in the absence of impurities.Therefore, decrease of TES with decreasing N seemsnatural.39,40

At high temperatures, the conductivity is due to thenearest-neighbor hopping. Its T-dependence is of activatedtype,

s5sAexp~2EA /T !, l,ls , ~10!

with the activation energy

EA;e2

kl ~11!

and the prefactor sA proportional to the probability of tun-neling between adjacent rods.

C. 3D systems with small impurity concentration

As impurity concentration decreases and l becomes largerthan ls , a number of dramatic changes appear in all keyquantities, such as the density of states, the localizationlength, and the effective dielectric constant. For example, aswe discuss in Sec. V, the dielectric constant starts to increaseexponentially with l because the polarizability of the crystalwith l.ls becomes limited not by the length l of individualchain segments but by the exponentially large length of Lar-kin domains. The soaring dielectric constant causes a rapiddrop of the ES parameter TES . In turn, this causes a collapseof the low-temperature resistivity in a narrow interval ls&l&lsln(W/T) ~see the descending branch of the curve in Fig.2!. Until this point, the notion that our system is opposite tothe conventional semiconductors, so that purer samples havehigher conductivities, seems to be working.

Once l exceeds lsln(W/T), the Larkin length Lx can betreated as effectively infinite. The VRH now involves hopsbetween low-energy states separated by distances shorterthan Lx . On such distances, the dispersion of the dielectricfunction becomes important. Each pair of low-energy chargeexcitations localized on their respective impurity clusters in-teracts via a strongly anisotropic electrostatic potential,which is not exponentially small only if the vector that con-nects the two charges is nearly parallel to the chain direction.Such an unusual interaction leads to a Coulomb gap that islinear in energy and independent of N, unlike the previouscase ~Sec. II B, l!ls), where the Coulomb gap is quadratic

FIG. 2. Logarithm of the resistivity as a function of the averageinterimpurity distance l51/Na

2 at a fixed temperature T!W . TheES, 2/5, Mott, and activation laws succeed each other with growingl.03541and N dependent. Another difference from Sec. II B is thatthe localization length jx for the tunneling in the chain di-rection is also independent of N,

jx;ls

Ars, ~12!

see Sec. V and Appendix B. This leads to a 2/5 law for theVRH conductivity

s5s0exp@2~T1 /T !2/5# , ~13!

where parameter T1, given by

T15C3e2rs

1/4

klsa

j, ~14!

does not depend on l and is, in this sense, universal. Here C3is yet another numerical coefficient of the order of unity. The2/5 law shows up as an intermediate resistivity plateau inFig. 2. This universal law for quasi-1D systems with l@ls isan analog of the universal ES law in isotropic systems.

We show in Sec. V that the Coulomb gap affects mainly afinite-energy interval uu&D , where D}gB can be called theCoulomb gap width. At larger energies, the density of statesof charge excitations coincides with the bare one, g().gB . Since gB is generated by impurity clusters whose con-centration diminishes with growing l, both gB and D de-crease with l. Eventually, the Coulomb gap becomes morenarrow than the range of energies around the Fermi levelresponsible for hopping at given T. At this point the Cou-lomb gap can be neglected and the 2/5 law is replaced by theconventional Mott law

s5s0exp@2~TM /T !1/4# , ~15!

where TM5C4 /gBjxj2

, C4;1. As l increases further, Tbeing fixed, s decreases because of diminishing gB . Thisgives rise to the ascending branch in Fig. 2. At such l, theelectron crystal behaves similar to a gapped insulator wherea lower impurity concentration corresponds to a lower carrierdensity, and thus, to a higher resistivity.

As l continues to grow, at some point the Mott VRHcrosses over to the nearest-neighbor hopping and shortly af-ter it becomes smaller than the conductivity due to thermallyactivated free solitons,

s5sAexp~2W/T !. ~16!

At even larger l, s ceases to depend on l, and so the impurityconcentration, see Fig. 2. Note that the activation energiesW;e2/kls @Eq. ~16!# and EA;e2/kl @Eq. ~11!# in Secs. II Aand II B, respectively, smoothly match at l;ls .

D. Summary of the regimes

The rich behavior of the conductivity as a function of land T is summarized in the form of a regime diagram in Fig.3. The 2/5 law applies in a broad range of l and T betweenthe ES and the Mott laws.

A convenient way to keep track of all the VRH exponentsderived in this paper is provided by Eq. ~17! below. We3-5

M. M. FOGLER, S. TEBER, AND B. I. SHKLOVSKII PHYSICAL REVIEW B 69, 035413 ~2004!would like to present it in a somewhat more general form,motivated by the following physical reasoning.

The diagram of Fig. 3 is obtained under the assumptionthat the tunneling in the transverse direction is not negligible(j is not too small!, so that the VRH has a 3D character.However, if the conducting chains are relatively distant fromeach other either along y or z direction or both, this conditionmay be violated. Examples of such systems are artificial ar-rays of quantum wires7 and carbon nanotubes.8 In those sys-tems, the 3D hopping is pushed to very lower temperatures,while at intermediate T the hopping can be either one or twodimensional. Generalizing the standard derivation of theVRH transport3,4 to the d-dimensional hopping and a power-law density of states g()}m, one obtains the conductivityin the form

s5s0exp@2~TVRH /T !l# , l5m11

m1d11 . ~17!

For d53 one recovers all the regimes discussed prior in thissection ~Mott, 2/5, and ES laws! by setting m successively to0, 1, and 2, according to the physical situation. For the sakeof completeness, the exponents for other ds in the samesituations are summarized in Table I. Inclusion of all suchregimes would transform Fig. 3 into a more complicated dia-gram, but would not change its general structure, so it willnot be shown here or discussed further below.

III. 1D SYSTEM

In the case of a single chain of electrons on a uniformpositive background @Fig. 1~a!# impurities divide the 1Delectron crystal in separate pieces, which behave as metallic

FIG. 3. Summary diagram for the transport regimes in a 3Dsystem. Domains of validity of ES @Eq. ~2!#, Mott @Eq. ~15!#, acti-vated @Eqs. ~10!, and ~16!# and 2/5 @Eq. ~13!# laws are shown.

TABLE I. The exponents l of VRH conductivity @Eq. ~17!# inthe cases of 3D, 2D, and 1D tunneling and a power-law dependentdensity of states g() that arises due to 3D Coulomb interactions.1D tunneling corresponds to j0.

Tunneling g5const g}uu g}2

3D 1/4 2/5 1/22D 1/3 1/2 3/51D 1/2 2/3 3/403541rods. The rod lengths x are distributed randomly around theaverage value l. Therefore, the background charge of a givenrod, Q52ex/a , is a random number. It can be written asQ52e(n1n), where n is an integer and n is a numberuniformly distributed in the interval 21/2,n,1/2.

In the ground state of the system each rod contains aninteger number nr of electrons, so that the rod has the totalcharge of q5e(nr2n2n). To find n we use the fact that theCoulomb self-energy of the rod is equal to q2/2Cr , whereCr5kx/@2 ln(x/a)# is the capacitance of this rod. On theother hand, the interaction of different rods does not containthe large logarithm ln(x/a), and can be neglected in the firstapproximation. Thus, the minimization of the total energy ofthe system amounts to minimizing the self-energies of therods. One can show then that, in the ground state, nr5n , sothat the charges of the rods are uniformly distributed in theinterval 2e/2,q,e/2. Indeed, if this is not true and thecharge of the given rod is q.e/2, then, by charge conjuga-tion symmetry, there should exist another rod with the samelength and the opposite charge 2q . Transferring an electronfrom the first rod to the second one we lower the total energyand make the absolute values of both charges smaller thane/2.

Below we use q52en and call rods with 0,n,1/2empty and with 21/2,n,0 occupied. We define the energyof an empty state, (x ,n), as the minimum work necessaryto bring to it an electron from a distant pure 1D electroncrystal with the same average linear density of electrons

~x ,n!5e2

kxlnS x

aD @~12n!22n2#5 e2kx lnS xa D ~122n!.

~18!

This energy is positive and vanishes only at n51/2. On theother hand, the energy of an occupied state is defined asminus the maximum work necessary to extract electron fromthis rod to the same distant pure 1D electron crystal. In thiscase the final result is identical to Eq. ~18! with n2n .Apparently, states with unu51/2 are exactly at the Fermilevel, which we take as the energy reference point ~it coin-cides with the electron chemical potential of a pure 1D crys-tal!. The low-energy states relevant to VRH transport haveunu21/2!1.

Now we can calculate the density of such states. Takinginto account the fact that this rod length x is distributed ac-cording to Poisson statistics, the disorder-averaged bare den-sity of states can be written as

gB~E !51l E0

1/2dnE

0

dxl expS 2 xl D dE2~x ,n!.

~19!

With a logarithmic accuracy, we can replace ln(x/a) byln(l/a). Then we use Eqs. ~18! and ~19! to find

gB~!51

20lF12S 11 0 D expS 2 0 D G , ~20!

where 05e2ln(l/a)/kl. We warn the reader that one shouldnot attribute much significance to the predictions of the3-6

VARIABLE-RANGE HOPPING IN QUASI-ONE- . . . PHYSICAL REVIEW B 69, 035413 ~2004!above formula in the region of high energies, *0, whereexcitations with charges larger than e will also contribute tovarious physical processes. On the other hand, close to theFermi level, for !0, only charge-e excitations are impor-tant, in which case Eq. ~20! is fully adequate, while gB isnearly constant,

gB.k

2e2ln~ l/a !. ~21!

For the calculation of the low-T transport, only this constantvalue is needed.

As mentioned in Sec. II, in 1D the (1/x)-Coulomb inter-action creates only marginal effects on the conductivity. Ifwe neglect them, in the first approximation, then standardMotts argument3 leads to the VRH that obeys the T1/2 law,Eq. ~2!, with TES5TM5C1 /gBjx , where jx is localizationlength for tunneling between distant rods. The value of jx isobtained from the following considerations. Tunnelingthrough an impurity that separates two adjacent rods can beviewed as a process in imaginary time that consists of thefollowing sequence of events.32 A unit charge assembles intoa compact soliton just before the impurity in one rod, tunnelsthrough the impurity, and finally spreads uniformly over theother rod. We assume that the chain is not screened by ex-ternal metallic gates. Then the tunneling probability can bewritten in the form exp(22s), where s is the dimensionlessaction13 s;rs

1/2ln3/2(l/a). Since s@1, tunneling paths withreturns can be neglected. Therefore, for electron tunnelingover distances x@l the action s should be multiplied by thenumber of impurities passed. The average number of suchimpurities is equal to x/l , which yields the total tunnelingprobability P}exp(22x/jx), with jx given by Eq. ~5!. With alogarithmic accuracy, the effect of the Coulomb gap is toreplace gB by g() evaluated at the characteristic hoppingenergy 5ATES(0)T , where TES(0) is defined by Eq. ~6!. Thelatter result follows from Eqs. ~21! and ~5!. The final expres-sion for the parameter TES is given by Eq. ~7!. We brieflynote that in a long enough chain clusters of atypicallydensely spaced impurities may exist. Tunneling through suchsegments would cost a higher tunneling action and thereforethe overall conductivity would be suppressed. In this paperwe assume that the chain is sufficiently short so that theserare clusters can be neglected, see a comment after Eq. ~7!.

Formulas ~6! and ~7! indicate that TES goes down as im-purity concentration N51/l decreases. This provides agradual crossover to the metallic behavior in a pure 1D sys-tem.

IV. 3D SYSTEM WITH A LARGE IMPURITYCONCENTRATION

In this section we consider a quasi-1D system made ofparallel chains which form a periodic array in the transversedirections. The chains are pinned by impurity centers, whichdivide them into metallic rods of average length l51/Na

2, where a

2 is the cross-sectional area per chain.The finite-size electron crystals in each rod are either com-pressed or stretched to accommodate an integral number of03541electrons, as in the case of a single chain. In this section weassume that for a typical rod, the energy of its longitudinaldeformation is smaller than the energy of its transverse cou-pling to rods on the neighboring chains. This can be the casewhen the periodic potential created by the neighboringchains is diminished because a is larger than a, and/orwhen the impurity concentration is large enough. Formally,the inequality a!l!ls needs to be satisfied, where ls is thesoliton length @Eq. ~1!#. Since metallic rods now completelyfill the 3D space @see Fig. 1~b!#, they modify the dielectricconstant of the system. As in the interrupted-strand model,38the dielectric constant is anisotropic. Along the chain direc-tion it has the value of

kx5k@11C5~ l/rD!2# , ~22!

where rD;a is the screening length, see Appendix. A, andC5;1 is a numerical constant.41 Transverse dielectric con-stants are unaffected, ky5kz5k . At large l, kx greatly ex-ceeds k , which leads to an anisotropic Coulomb interactionin the form42

U~r !5e2

kAx21~kx /k!r2, ~23!

where r2 5y21z2. In spite of the large dielectric constant,

the Coulomb interaction is long range and thus creates aCoulomb gap. Using the standard ES argument,4,43 the fol-lowing density of states of charge excitations is obtained

g~!53p

k2kx

e62. ~24!

It differs from the conventional formula for an isotropic me-dium only by the presence of kx instead of k . In order tocalculate the VRH conductivity we still have to discuss thetunneling probability. It is important that Eq. ~23! holds onlyat x@l . The interaction between charge fluctuations on thesame rod is short range due to screening by neighboringchains.44 Tunneling along the x axis takes place similarly tothe case of a single chain, but screening of the Coulombinteraction leads to smaller action of the order of s;Arsln(l/a). Therefore, for the localization length in the xdirection we get

jx5ls

;l

Arsln~ l/a !. ~25!

The tunneling in the y and z directions is accomplished bysingle-electron-like excitations. The probability of tunnelingdecays exponentially, P}exp(22r /j), at large transversedistances r . Here j is the transverse localization lengthgiven by Eq. ~8! and discussed in more detail in Appendix B.We assume that j is not vanishingly small compared to jx ,in which case the VRH has a 3D character and can be cal-culated by the percolation approach ~see Ref. 4, and refer-ences therein!.45 This calculation differs from the isotropiccase by the replacement of the isotropic dielectric constant k3-7

M. M. FOGLER, S. TEBER, AND B. I. SHKLOVSKII PHYSICAL REVIEW B 69, 035413 ~2004!and the isotropic localization length j by their geometricaverages over the three spatial directions:

TES5C3e2

~k2kx!1/3

1

~jxj2 !1/3

. ~26!

With the help of Eqs. ~22! and ~25! the expression for TESreduces to Eq. ~9!, where the screening length of the electroncrystal has been taken as rD;a and numerical coefficientsabsorbed in C3.

V. 3D SYSTEM WITH A SMALL IMPURITYCONCENTRATION

In this section we study the crystal pinned by impuritieswith a low concentration N so that the condition l@ls issatisfied. As discussed in Sec. I all key quantitiesdensity ofstates, localization length, the screening of Coulombinteractionsundergo dramatic changes compared to thecase of high impurity concentration. We address suchchanges in the three separate sections below.

A. Pinning of a quasi-1D crystal by strong dilute impurities

In this section we study the ground-state structure andscreening properties of the crystal with low impurity concen-tration.

We start by reviewing the physical meaning of ls given byEq. ~1!, in which a5Y /Y x is the anisotropy parameter, andY x , Y parametrize the energy of an elastic distortion of thecrystal,

Eel512E d3r@Y x~]xu !21Y~u !2# . ~27!

As shown in Appendix A, at large rs the longitudinal elasticmodulus Y x is dominated by Coulomb effects,

Y x;e2/ka2a2

. ~28!

The transverse modulus Y can be substantially smaller thanY x even when the ratio a /a is only modestly large. ForCoulomb interaction, Y}exp(22pa /a).

The energy Eel in Eq. ~27! is essentially the short-rangepart of the Coulomb energy. The total energy also includesthe long-range Coulomb part ~see Appendix A! and the pin-ning part ~see Appendix C and below!. Strictly speaking, Eq.~27! is valid only for small gradients of the elastic displace-ment field u(r); however, it can be used for order-of-magnitude estimates down to microscopic scales r;a andx;ls . In this manner, one can derive formula ~1! by mini-mizing Eel under the condition that u changes from 0 to aover a segment of length ls on a single chain. For moredetails, see Refs. 27, 23, and Appendix A.

The inequality l@ls imposes the upper limit on the impu-rity density:

N!Aa/a3 . ~29!03541Below we show that at such N the ground state of the elec-tron crystal is determined by an interplay of individual andcollective pinning.46

Without impurities the crystal would have a perfect peri-odicity and long-range 3D order. Impurities cause elastic dis-tortions of the lattice. The strongest distortions, of dipolartype, are localized in the vicinity of impurities, see Sec. I andFig. 1~c!. Such dipoles have a characteristic size ls ~same asfree solitons!, are well separated from each other, and occupyonly a small fraction of the space. Their creation is advanta-geous because the associated energy cost ~elastic plus Cou-lomb! is of the order of W;e2/kls per impurity, whereas theenergy gain is a much larger electron-impurity interactionenergy 2e2/ka . This is the essence of individual ~strong!pinning phenomenon, which provides the dominant part ofthe pinning energy density. The collective ~weak! pinningresults from interaction between the dipoles. Let us demon-strate that such interaction cannot be neglected at sufficientlylarge length scales. By solving the elasticity theory equations~generalized to include the Coulomb interactions!, it can beshown27 that a dipolar distortion centered at a point (xi ,ri)has long-range tails that decay rather slowly with distance,u;AiarD /Aaux2xiu. This displacement is confined mainlywithin a paraboloid ur2riu2&AarDux2xiu. Note that thesegment 0,x,xmin[a(l/AarD)1/2 of the paraboloid,

r2 5AarDuxu, ~30!

contains on average one impurity. Parameter rD;a , whichwe already encountered in Sec. IV, has the meaning of thescreening length. It is related to the longitudinal elasticmodulus Y x as follows:

rD2 5

k

4pe2a2a

4 Y x , ~31!

see Appendix A.If, in the first approximation, we choose to neglect the

interaction among the dipoles, then we should simply addtheir far elastic fields treating the amplitudes 21&Ai&1 asrandom variables. We immediately discover the logarithmicgrowth of u with distance,

^@u ~x ,0 !2u ~0,0 !#2&;NExmin

x

dx8E0

AarDx8dr2 S arDAax8D

2

5S a2C6D S lsl D lnS xxminD , ~32!where the bar over u indicates that we refer to the value of uaway from the immediate vicinity of a dipole and ^&stands for disorder averaging.

The logarithmic growth of the elastic displacements waspreviously derived for the model of weak pinning centers inearly works21,22 on the subject. In those calculations the nu-merical coefficient C6 is large and is inversely proportionalto the impurity strength. In our case C6;1. Apart from that,Eq. ~32! demonstrates that the case of strong pinning centersis essentially similar. Therefore, as customary for weak pin-3-8

VARIABLE-RANGE HOPPING IN QUASI-ONE- . . . PHYSICAL REVIEW B 69, 035413 ~2004!ning models we define the longitudinal Lx and transverse LLarkin lengths as the length scales where ^Du2&[^@u(r)2u(0)#2&;a2. From Eqs. ~30! and ~32! we obtain

Lx5xminexp~C6l/ls!, L5rminexp~C6l/2ls!, ~33!

where rmin5(AarDxmin)1/2.

Alternative derivation of Eqs. ~32! and ~33! based on en-ergy estimates is given in Appendix C. It elucidates that theslow logarithmic growth of ^Du2& is rooted in the importantrole of long-range Coulomb interaction in the elastic re-sponse of a quasi-1D crystal. An isotropic electronic crystaladjusts to pinning centers primarily by means of sheardeformations47 that do not cost much Coulomb energy. As aresult, in the isotropic crystal Du grows algebraically withdistance. In contrast, in quasi-1D crystals and CDW, wherethe elastic displacement is a scalar ~electrons move onlyalong the chains!, no separate shear deformations exist. Thebuildup of the Coulomb energy that accompanies longitudi-nal compressions translates into an exceptionally large rigid-ity of the electron lattice and exponentially large Lx and L .

At distances exceeding the Larkin lengths the dipoles canno longer be treated as independent. Indeed, the energy costEs of a given dipole is determined by the minimal distanceby which the crystal has to distort to align an electron withthe impurity position. Therefore, just like the energy of a rodin the previous sections, Es has a periodic dependence onn5$(xi2u )/a%, where $% denotes the fractional part. Esvanishes at n50 and reaches a maximum value of ;W atn561/2. Therefore, as soon as the cumulative effect ofother dipoles attempts to elevate unu above 1/2, a 2p-phaseslip should occur to adjust Es(u ) to a lower value. The over-all effect of such adjustments is to enhance the pinning en-ergy. This additional energy gain can be viewed as the col-lective pinning effect. Using standard arguments ~seeAppendix C!, we relate the extra pinning energy density tothe Larkin length

Epin;2W/a2 Lx;2Y~a/L!2. ~34!We will now use this result to estimate the asymptotic value

kx[e~qx0,q50 ! ~35!of the longitudinal component of the dielectric function.

Without impurities, the dielectric function has the form

e~q!5k1qx

2

q2krD

22

qx21aq

2 , ~36!

see Appendix A and, e.g., Ref. 48. Based on previouswork49,22,10 we assume that a reasonable description of di-electric screening in a system with impurities is obtained ifwe replace the random distribution of pinning centers by acommensurate pinning with the same Epin . In this case wecan use the concept of the dielectric function even for thedisordered system. To derive the modified expression for thedielectric function, one can add the term Epin(u/a)2 to theright-hand side of Eq. ~A2! in Appendix A and repeat thesteps outlined thence. It is easy to see that the net effect of03541~commensurate! pinning is to augment the combination qx2

1aq2 ~proportional to the elastic resorting force! by the

term 2Epin /Ya2, which comes from the additional restor-ing force due to impurities. In this manner we obtain

e~q!5k1qx

2

q2krD

22

qx21a~q

2 1L22!

, ~37!

kx5k

a S LrD D2

5k

AaLxrD

;expS C6 llsD . ~38!However, for our purposes a cruder approximation will besufficient, namely, we can assume that at distances shorterthan the Larkin length, the system screens as though it is freeof impurities, Eq. ~36!; at distances larger than the Larkinlength, the dielectric function is replaced by a constant, Eq.~38!. In the following section we will use Eqs. ~36! and ~38!to derive the functional form of the Coulomb gap in theregime l@ls .

B. Bare density of states and the Coulomb gap at lowimpurity concentration

In order to describe the low-T transport at low impurityconcentration, l@ls , we need to determine the origin of low-energy charge excitations in this regime. This poses a con-ceptual problem. Indeed, such excitations do not exist in thebulk ~away from impurities! where the creation energy ofcharge-e excitations is bounded from below by the energy Wof a 2p soliton. At first glance, the impurities do not helpeither. As mentioned in Sec. I, near isolated impurities thereis an energy gap for charge excitations, which is not muchsmaller than W. This is because a single impurity appreciablydisturbs the crystal only within the region of length ls . Thedisturbance is electrically neutral ~dipolar! in the groundstate. Creation of a charge-e excitation near such an impurityrequires an energy of the order of e2/kls;W . Let us nowshow that charge excitations of arbitrary low energies never-theless exist. They come from impurity clusters. Each clusteris a group of a few impurities spaced by distances of theorder of ls or smaller. ~It can be viewed as a microscopicinclusion of the l&ls phase.! Below we demonstrate that theclusters provide the bare density of states at zero energy,which decreases with l no faster than a power law,

gB5k

e2a2 S lsl D

b12

, ~39!

where exponent b is of the order of unity and is independentof l. The calculation of b has to be done numerically, whichwe leave for future work.

To prove that Eq. ~39! gives the lower bound on gB ,consider the configuration of 2M12 impurities shown inFig. 4. The impurities define a cluster of 2M11 short rods,each of approximately the same length c!ls . The cluster isflanked by two semi-infinite segments at the ends. We as-sume that M is sufficiently large so that L5Mc is muchgreater than ls . Suppose that the length of the central rod in3-9

M. M. FOGLER, S. TEBER, AND B. I. SHKLOVSKII PHYSICAL REVIEW B 69, 035413 ~2004!units of a is close to a half integer, so that the charge of thisrod is restricted to the set of values q5(21/22d1n)e ,where n is an integer and 0,d!1. For the low-energy stateswe only need to consider two possibilities, q5(21/22d)eand (1/22d)e . The lengths of the other short rods in ourconstruction are chosen to be close to integer multiples of a.Then those rods can be considered charge neutral. Finally,we assume that the position of the leftmost impurity restrictsthe charge of the left semi-infinite cluster to 1/41dL1nL ,where udLu!1 and nL is another integer. Under these as-sumptions, the charge of the right semi-infinite segment isfixed to the set 1/41d2dL1nR .

One possible candidate for the ground state is the chargeconfiguration shown in Fig. 4~a!. It is overall neutral, has thecharge of the central rod equal to q5(21/22d)e , and thetotal energy

Ea52

Lke

4 S 2 e2 D1 12Lk S e4 D2

1e2~1/21d!2

2Cr12W1/4

52732

e2

Lk 1e2

2Cr~1/21d!212W1/4 . ~40!

Here we made use of the conditions L@ls , d!1. W1/4 de-notes the self-energies of the semi-infinite segments ~whichcontain 1/4 solitons!, and Cr;kc is the self-capacitanceof the central rod. If the configuration shown in Fig. 4~a! isindeed a ground state, it has an unusual feature that thecharge of the central rod exceeds 1/2 by the absolute value,contrary to the arguments in Sec. III. To establish the condi-tions for the cluster to have such a ground state, we need tocompare Ea with energies of other possible states. Of those,some are neutral and some are charged. One competing neu-tral state is shown in Fig. 4~b!. It has energy

Eb521132

e2

Lk 1e2

2Cr~1/22d!21W1/41W3/4 , ~41!

where W3/4 is the energy of the 3/4 soliton on the right ofthe cluster in Fig. 4~b!. For the energy difference we have

FIG. 4. An example of an impurity cluster that provides low-energy charge excitations at l@ls . ~a! Distribution of charges in theground state. For simplicity, we chose dL5d/2. The cluster is elec-trically neutral. ~b! Competing neutral state. ~c! Excited state ob-tained by increasing the charge of the central rod by one unit.035413Eb2Ea5e2

Crd2

18

e2

Lk 1W3/42W1/4 , ~42!

so that the state shown in Fig. 4~a! wins if

d,Cre2

F ~W3/42W1/4!2 18 e2

LkG . ~43!Since W3/42W1/4;W;e2/kls and L@ls , the right-handside of the inequality ~43! is positive, and so d.0 that sat-isfy such an inequality do exist. The only other viable com-peting neutral state is similar to that shown in Fig. 4~b! ex-cept the 3/4 soliton is formed on the left semi-infinitesegment. The energy of that state is also Eb , and so it doesnot lead to any further restrictions on d .

Now let us examine the charged states. There is only oneviable competitor, with q5(1/22d)e , as shown in Fig.4~c!. If the energy difference 5Ec2Ea is positive, thenFig. 4~a! represents the true ground state and gives thecreation energy of the charge-e excitation. If is negative,then the ground state is as shown in Fig. 4~c!, while thecharge and the energy of the lowest-energy charge excitationare equal to 2e and 2 , respectively. An elementary calcu-lation yields

52e2

Crd1

e2

2kL , ~44!

from which we conclude that it is possible to obtain arbitrarysmall of both signs by tuning d sufficiently close toCr /2Lk;c/L , without violating the inequality ~43!. Thisproves that clusters are able to produce a finite gB(0). In theground state some of these clusters are neutral ~empty!and some are charged ~occupied!.

Now let us try to estimate gB due to clusters. In the aboveargument we required strong inequalities c!ls and L@ls toprove the existence of a nonzero gB with mathematical rigor.Physically, it seems reasonable that such inequalities can besoftened to c&ls and L*ls , in which case d&1 and M;1, i.e., only a few ~of the order of unity! impurities areneeded. It is also clear that the rods to the left and to the rightof the central one need not be exactly neutral for the argu-ment to go through. Then the probability of forming the de-sired cluster is comparable to the probability of finding 2M12 ~a few! impurities on the same chain with nearest-neighbor separation less than ls . Assuming that impurity po-sitions are totally random and independent, we obtain theestimate of Eq. ~39!.

Now let us calculate the form of the Coulomb gap in theactual density of states g(). At exponentially small ener-gies, !

*5e2/kLx , the Coulomb gap is determined by

the interactions on distances exceeding the size of the Larkindomain. At such distances the interaction has the form ~23!,leading to the usual parabolic Coulomb gap given by Eqs.~24! and ~38!. We put these equations side by side below forthe ease of reading,

g~!53p

k2kx

e62, !

*, ~45!-10

VARIABLE-RANGE HOPPING IN QUASI-ONE- . . . PHYSICAL REVIEW B 69, 035413 ~2004!*

5e2

kLx5

e2

AarD

1kx

, ~46!

kx;expS C6 llsD . ~47!To ascertain the region of validity of Eq. ~45! one needs tomake sure that g() does not exceed the bare density ofstates gB . It is easy to see that this is the case here. At thelargest energy ;

*@Eq. ~46!#, g() is exponentially small,

g}exp(2C6l/ls), and so it is indeed much smaller than gB;(ls /l)b12. In this sense gB is large enough to ensure thevalidity of the parabolic law over the full range of specifiedin Eq. ~45!.

At larger than *

, the Coulomb gap is governed byinteractions within the volume of a Larkin domain and thedispersion of the dielectric function e(q) becomes important.The interaction potential is defined by U (q)54pe2/e(q)q2and Eq. ~37! in the q space. In real space, it is given by27

U~r!.e2

2kuxu expS 2 14Aa r2

rDuxu D , ~48!a!r!min$Aauxu,L%,

rD

Aa!x!Lx . ~49!

The potential U(r) is appreciable only within the paraboloiddefined by Eq. ~30!, where it behaves as U(r).e2/2kuxu. Amore precise statement is that the surface of a constant U isa paraboloidlike region

r2 .4AarDuxulnS e22Ukuxu D . ~50!

For U@*

, this surface belongs to the domain ~49! whereEq. ~48! holds.

To calculate the functional form of the Coulomb gap weuse the self-consistent mean-field approximation due toEfros,43 according to which g is the solution of the integralequation

g~!5gBexpF2 12E0Wd8g~8!V~18!G , ~51!where V(U) is the volume enclosed by the constant-U sur-face ~50!,

V~U !.pAa

4e4rD

k2U2. ~52!

Equation ~51! follows from the requirement that the groundstate must be stable againts a transfer of a unit charge froman occupied state with energy 28 to an empty state withenergy . Such a stability criterion28,4 can be expressed bymeans of the inequality 182U(r).0, where r is thevector that connects the two sites. Thus, the integral on theright-hand side of Eq. ~51! counts the pairs of states thatwould violate the stability criterion if positioned randomly.035413The Coulomb gap can be viewed as the reduced statisticalweight of stable ground-state charge distributions with re-spect to that of the totally uncorrelated ones.

The solution of Eq. ~51! has the asymptotic form

g~!.8

pAak2

rDe4 uu, *!uu!D . ~53!

We see that the unusual interaction potential of Eq. ~48!leads to a nonstandard Coulomb gap, which is linear in a 3Dsystem. The weaker ~linear instead of the standard quadratic!suppression of g is due to a metallic screening of the Cou-lomb potential in the major fraction of solid angle. The en-ergy scale D in Eq. ~53! is defined by the relation g(D);gB . Using Eq. ~39!, we can estimate D as follows:

D5AarD

k2e4gB;WS lsl D

b12

. ~54!

At @D , the solution of Eq. ~51! approaches the bare den-sity of states, g.gB , and so D has the meaning of the Cou-lomb gap width. On the lower-energy side, at ;

*, the

linear Coulomb dependence of Eq. ~53! smoothly matchesthe quadratic dependence of Eq. ~45!. All such dependenciesare summarized in Fig. 5.

C. VRH transport in a system with small impurityconcentration

The VRH transport at l@ls involves quantum tunneling ofcharge excitations between rare impurity clusters. In Appen-dix B we advance arguments that the charge tunnels alongthe chains in the form of 2p solitons and derive the follow-ing estimate for the corresponding longitudinal localizationlength:

jx;ls

ArsF11 lsl lnS lsa D G

21

, l.ls . ~55!

At l@ls this formula goes over to Eq. ~12!, while at l;ls itsmoothly matches with Eq. ~25!. As for the interchain tun-neling, it is still accomplished by single-electron-like excita-

FIG. 5. The density of states of charge excitations in a 3D sys-tem with l@ls . The parabolic Coulomb gap at low energies issucceeded by the linear Coulomb gap, then by the bare density ofstates gB created by impurity clusters.-11

M. M. FOGLER, S. TEBER, AND B. I. SHKLOVSKII PHYSICAL REVIEW B 69, 035413 ~2004!tions and the corresponding localization length j is stillgiven by Eq. ~8!, see Appendix B. We will assume thatj /a is not vanishingly small, in which case the VRH hasthe 3D character.45

The density of states and localization lengths is all theinformation we need to calculate the VRH conductivity. Amore complicated dependence of the density of states g()on energy in the case at hand, l.ls , brings about a largervariety of possible transport regimes.

At the lowest temperatures we still have the ES law withparameter TES given by Eq. ~26!. Due to the exponentialgrowth of the longitudinal dielectric constant kx with l @Eq.~38!#, TES decreases exponentially, TES}exp(2C6l/3ls). Thisdependence entails a precipitous drop of 2ln s as a functionof l, as soon as l exceeds ls , see Fig. 2. Such an enhancementof the conductivity is due to progressively more efficientscreening of the long-range Coulomb interactions ~steeplyincreasing kx), which enhances the density of states insidethe Coulomb gap, see Eq. ~45!.

The ES law ~2! holds until the range of s that contributeto the VRH transport,4 uu&(TTES)1/2, fits inside the para-bolic part of the Coulomb gap, uu&

*. For a fixed T!W

this gives the condition l&lsln(W/T).At larger l, the unusual linear Coulomb gap ~53! leads to

the 2/5 law for the VRH transport @Eq. ~13!#, which we re-produce below for convenience,

s5s0exp@2~T1 /T !2/5# . ~56!

As emphasized in Sec. II, parameter T1 is impurity indepen-dent and is, in this sense, universal, see Eq. ~14!. This be-havior leads to the intermediate plateau at the graph in Fig. 2.

The range of energies that contributes to the VRH in the2/5-law regime is given by uu&(T1 /T)2/5T . At l;ls(W/T)g, where g53/@5(b12)#&0.3, this range be-comes broader than the Coulomb gap width D @Eq. ~54!#. Atsuch and larger l, the Coulomb gap can be neglected, and theVRH begins to follow the usual Mott law @Eq. ~15!#

s5s0exp@2~TM /T !1/4# , ~57!

with parameter TM increasing with l according to TM}1/gB}(l/ls)b12. The growing TM leads to exponentiallyincreasing resistivity, represented by the ascending branch ofthe curve in Fig. 2. Physically, the suppression of the dcconductivity stems from decreasing density of low-energystates available for transport, just like in conventional dopedsemiconductors3,4 or commensurate CDW systems.39

VI. CONCLUSIONS AND COMPARISON WITHEXPERIMENT

It is widely recognized that interactions must play a sig-nificant role in determining the properties of 1D andquasi-1D conductors, because in such materials the dimen-sionless strength of the Coulomb interaction is very large,rs@1. In the presence of impurities, these systems behave asinsulators and do not possess metallic screening. Thus, theinteractions are both strong and long range. Our main goal inthis paper was to understand the effect of such interactions035413on the nature of the low-energy charge excitations and theirOhmic dc transport. To that end we formulated a genericmodel of an anisotropic electron system with strong Cou-lomb interactions and disorder and presented its theoreticalanalysis. We elucidated the origin of the low-energy chargeexcitations in this model and demonstrated that their densityof states possesses a soft Coulomb gap. In 3D case, we foundthat the Coulomb gap exhibits a power-law dependence onthe energy distance from the Fermi level. We discussed howthe prefactor and the exponents of this power law vary as afunction of the impurity concentration and other parametersof the model. We also discussed how the Coulomb gap ismanifested in the variable-range hopping conductivity at lowtemperatures.

One of the central results of our theory is a nonmonotonicdependence of s on the impurity concentration N, as shownin Fig. 2, where we sketched s as a function of l51/Na

2,

i.e., as N decreases, at fixed T. As clear from that figure andthe discussion in Sec. II, at large N the conductivity increasesas N decreases, similar to behavior found in metals. In con-trast, at small N the conductivity drops as N goes down,which resembles the behavior of doped semiconductors. Forintermediate N, our theory predicts the existence of anN-independent plateau.

Another way to represent these theoretical predictions,common in semiconductor physics, is shown in Fig. 6. Inthat figure the dependence of the logarithm of conductivity

FIG. 6. Logarithm of the conductivity vs the inverse tempera-ture for several samples labeled in the order of increasing l, i.e.,sample purity. ~a! Curve 1 corresponds to l5l1,ls and displaysonly the ES regime. The higher-T activation regime is beyond thelimits of the graph. Curve 2 is for l5l2*ls , and so the both theactivation behavior and the ES law are visible. Curve 3 is for l5l3*l2. It shows the complete sequence of the transport regimes:the activation, Mott, 2/5, and ES laws. Curve 4 depicts the behaviorof a sample with an impurity concentration another notch lowerthan that of sample 3. In panel ~a! it goes through the activation andthe Mott regimes. In panel ~b! that covers considerably lower T,curve 4 also exhibits the 2/5 law, followed by the ES law. The 2/5law is shown by the solid line along the lower edge of the shadedregion. The curves can skim along this line but cannot cross it. Thehigher the sample purity the lower the temperature at which thesample starts to exhibit the 2/5 law but also the wider the range ofT over which this law persists. Curve 5 depicts the case l@ls whereany kind of VRH transport corresponds to Ohmic resistances higherthan the experimental measurement limit, so that only the activatedtransport can be observed.-12

VARIABLE-RANGE HOPPING IN QUASI-ONE- . . . PHYSICAL REVIEW B 69, 035413 ~2004!on the inverse temperature is depicted for a series ofsamples, each with fixed N. An unusual circumstance illus-trated by Fig. 6 is the crossing of the curves that correspondto different samples. In Fig. 6~a! we show that up to twocrossing points may exist between one curve that corre-sponds to l,ls ~curve 2! and another curve for l.ls ~curve4!. The higher-T crossing occurs when the curve 4 goesthrough the activation regime, the lower-T onewhen it ex-hibits the Mott VRH. The dirtier (l,ls) sample obeys the ESbehavior in both instances. Another property we tried to em-phasize in Fig. 6 is the role of the 2/5 law as the upper boundof the conductivity regardless of the sample purity. Forsamples with low impurity concentration the 2/5 law is alsothe envelope curve, see Fig. 6~b!.

Let us now turn to the experimental situation. The trans-port behavior of a number of organic compounds, includingTMTSF-DMTCNQ, TTF-TCNQ, and NMe-4-MePy(TCNQ)2, is indeed in a qualitative agreement with ourtheory. It should be clarified that such materials form CDWphases that in addition to the usual 2kF periodicity, have anappreciable or, in some cases, even predominant 4kF har-monics. The latter is considered the evidence for the strongCoulomb interaction,11 and so is precisely the case we stud-ied in this paper.

In the experiments of Zuppiroli et al.50,35 the transport inthe aforementioned compounds was studied as a function ofdefect concentration, which was varied in situ by irradiationof samples by high-energy particles. Admittedly, the natureof the such defects is not known with certainty. Some sug-gestions in the literature include atomic displacements, bro-ken bonds, polymer cross linking, and charged radicals. Atthe same time, the effect of the irradiation on transport seemnot to depend much on the type of particles used ~x rays,neutrons, or electrons! and instead to correlate primarily withthe total absorbed energy.35 This fact is interpreted as evi-dence that microscopically different defects influence thetransport in electron crystals in a similar way, so long as theyact as strong localized pinning centers. Under this assump-tion, it is legitimate to compare the data from the irradiationexperiments with our theory even though so far we assumedthat defects are created by charged acceptors ~see Sec. I!. Wedo so in some detail below.

In Fig. 7 we show an extensive set of data on transport inirradiated TMTSF-DMTCNQ that we assembled by digitiz-ing Fig. 2 of a review paper by Zuppiroli,35 and originalreferences therein. Apparently, some data series in this figurerepresent the same sample with successively increasing ra-diation dose, and some correspond to physically differentspecimens. For simplicity, we refer to all of them as differentsamples. The percentage labels on the plot are the estimatesof the molar concentrations of defects c quoted by the ex-perimentalists. The points on the c50 trace are from anunirradiated sample.

As shown in Fig. 7, the data for the two most disorderedsamples can be successfully fitted to the activation law andthe next two samplesto the ES formula. This transitionfrom the activation to the ES law with increasing disorder isin agreement with our theory ~cf. curves 1 and 2 in Fig. 6!.The obtained fit parameters EA and TES are given in Table II.035413Both EA and TES scale roughly linearly c, in agreement withEqs. ~11! and ~9!. From Eq. ~10! we deduce that klc5ka;1 nm, which has the correct order of magnitude ~assumingk;1). One should keep in mind here that the absolute num-bers for c were obtained by the authors of Ref. 51 usingcertain arguable assumptions. In our opinion, the scalingwith c may be more reliable than the absolute values quotedbecause ~if no annealing occurs! the relative magnitude of cshould scale linearly with the irradiation time, known to ex-perimentalists without any fitting parameters. CombiningEqs. ~9! and ~11! we further deduce that a /j and l/jxratios are some modest numbers less than ten, as may beexpected from Eqs. ~25! and ~8!.

As a final remark on high-disorder samples, we wouldlike to mention that the scaling of the longitudinal dielectricconstant kx with the defect concentration ~irradiation time!consistent with Eq. ~22! was reported in a separate set ofexperiments on Qn~TCNQ! by Janossy et al.52 Together withthe transport data, this makes a compelling case for the va-lidity of the metallic-rod ~interrupted-strand! model for or-ganic electron crystals with l!ls . For such systems we canclaim a semiquantitative agreement with the experiment.

Let us now turn to the conductivity of weakly damagedsamples.35 As one can see from Fig. 7 they show metallicbehavior at high T, a conductivity maximum at the Peierlstemperature of about 42 K, and a decrease in s , i.e., the

FIG. 7. Low-temperature conductivity of TMTSF-DMTCNQsamples damaged by x-ray radiation. The dots on c>1.9% and c

M. M. FOGLER, S. TEBER, AND B. I. SHKLOVSKII PHYSICAL REVIEW B 69, 035413 ~2004!semiconducting behavior, at lower T. As T drops by a factorof 2 or so below the Peierls temperature, the decrease of swith 1/T becomes considerably more gentle than an activa-tion law. One cannot help noticing a similarity between thebehavior of c51.9%,0.35%, and 0% samples in Fig. 7 andcurves 1, 3, and 4, respectively, in Fig. 6. There is also anunambiguous evidence for the existence of the crossing pointbetween, e.g., c50% and c1.2% traces at T521 K ~seebelow!. However, an attempt to fit the c50% data to theMott law is not particularly successful, see Fig. 7. Therefore,we only wish to emphasize a qualitative agreement with ourprediction that for a fixed T, the conductivity of clean anddirty samples should show opposite trends, see Figs. 2 and6. Indeed, the conductivity of the low-disorder samples (c1.9%)where it decreases. In fact, another experiment showed thiscontrasting behavior in a great detail. In that experiment,51ln s was measured at the fixed temperature of T521 K,while defect concentration was varied essentially continu-ously over the range of 0%,c,2.5%. It was found that sinitially increases by two orders of magnitude, reaches amaximum, and then drops by five orders of magnitude as afunction of c. Overall, this is in a qualitative agreement withFig. 2 except instead of the well-defined 2/5 plateau, ln sshows only a broad maximum. Similar features are demon-strated also by TTF-TCNQ and NMe-4-MePy ~TCNQ! 2, seeFig. 1 in Ref. 50, and Figs. 11 and 12 in Ref. 35. We con-clude that for low-disorder CDW organics our theory agreeswith experiment in some gross qualitative features. Thequantitative agreement cannot be verified because the dy-namical range of measured conductivities is too narrow. Fur-ther low-temperature experiments are desired to clarify thesituation and to prove or disprove the existence of the 2/5law.

Let us now switch to inorganic CDW. Several commentsare in order. The electron-electron interactions in these ma-terials are also very strong,10,53 rs;100. However, inorganicCDW are predominantly 2kF , and there is an ample evi-dence for the important role of the electron-phonon couplingin the CDW dynamics. This coupling can lead to an enhance-ment of the electron effective mass49,10 that would result in ashort localization length. If the mass enhancement is indeedlarge, the VRH transport should be observable only in mate-rials with short hopping distances, i.e., large impurity con-centrations. Examples include highly doped bronzes34 andperhaps, the Pt-chain compound KCP.54,55 From this perspec-tive, the reports of a VRH-like transport in a relatively puresamples of TaS3 @Refs. 56 and 57# and blue bronze @Refs. 58and 59# are puzzling and require further investigations.

Finally, let us comment on another broad class ofquasi-1D systems, conducting polymers. In comparison toCDW, polymers have a much higher degree of structural dis-order and a complex morphology that depends on the prepa-ration method. Typical samples contain a mixture of crystal-line and amorphous regions, with the correlation length60 ofthe order of 10 nm. In the undoped state polymers are com-mensurate CDW semiconductors with a Peierls-Mott energygap2 ;1 eV. Doping shifts these systems away from com-035413mensurability point and suppresses the gap but it is ofteninhomogeneous and is an additional source of disorder. Atlow and moderate doping the T dependence of the conduc-tivity often resembles ES and/or Mott VRH laws, see a shortreview in Ref. 37. A systematic study of the VRH conduc-tivity dependence on doping has been attempted by Aleshinet al.61 In those experiments doping of poly~3,4-ethilenedioxythiophene/poly~sterenesulfonate! ~PEDT/PSS!samples was varied by controlling the pH of the solution atthe sample preparation stage. It was observed that at pH,4 the VRH exponent l @Eq. ~17!# was close to 0.5 andTVRH decreased with pH, while at larger pH, l was close to0.452/5 and TVRH did not depend on pH. This resembles thebehavior that follows from our theory, provided the concen-tration of the pinning centers decreases with pH. We leavethe tasks of extending our theory to the case of conductingpolymers and explaining these intriguing experimental re-sults for future investigations.

ACKNOWLEDGMENTS

This work was financially supported by UCSD ~M.M.F.!,NSF Grant No. DMR-9985785 ~B.I.S.!, and INTAS GrantNo. 2212 ~S.T.!. We thank S. Brazovskii, A. Larkin, and S.Matveenko for valuable comments, K. Biljakovic, O. Chau-vet, S. Ravy, D. Staresinic, and R. Thorne for discussions ofthe experiments, and Aspen Center for Physics, where a partof this work was conducted, for hospitality.

APPENDIX A: DIELECTRIC FUNCTION ANDELASTICITY IN A CLEAN QUASI-1D ELECTRON

CRYSTAL

In this section we derive expressions for the elasticmoduli and the dielectric function of a pure crystal.

Following the literature on interacting 1D electronsystems16 and CDW,49 we describe dynamics of electrons onith chain by a bosonic phase field w i(x ,t). The long-wavelength components of electron density ni is related to w iby ni5]xw i /2p . The elasticity theory of the system can beformulated by identifying the elastic displacement u with(2p/a)w and taking the continuum limit. Neglecting weakinterchain tunneling and dynamical effects we choose ourstarting effective Hamiltonian H in the form

H5E dx@H01HC# , ~A1!H05

12 Cx

0(i

ni21(

i jJ i jcos~w i2w j!, ~A2!

HC512 (i j E dxdx8ni~x !Ui j~x2x8!n j~x8!. ~A3!

Let us briefly describe the notations here. The Hamiltonian issplit into the short-range (H0) and the long-range Coulomb(HC) parts. Ji j represents the Coulomb coupling between theCDW modulations of electron density on chains i and j. Cx0 isthe charge compressibility of a single isolated chain. In a-14

VARIABLE-RANGE HOPPING IN QUASI-ONE- . . . PHYSICAL REVIEW B 69, 035413 ~2004!large-rs 1D system Cx0 is dominated by the exchange-

correlation effects, leading to62

Cx0;2

2e2

kln

a

R , ~A4!

where R is the characteristic radius of the electron chargeform factor in the transverse direction, i.e., the radius ofthe chain. In most physical realizations, we expect R!a andnegative Cx

0. The positivity of the elastic modulus of the

system, required for thermodynamic stability, is recoveredonce we take into account the long-range part HC of theinteraction energy, parametrized by the kernels Ui j . Ui j isdefined as the bare Coulomb kernel U0(r)5e2/kr convo-luted with the single-chain form factors F, e.g., F(q)5exp(2q2 R2).

We are interested in a linear response where the cosines inEq. ~A2! can be expanded in w thereupon the effectiveHamiltonian becomes quadratic. If an external electrostaticpotential V ext(q) acts on the system, the total equilibriumpotential V (q) will in general contain Fourier harmonicswith wave vectors q1G, where G are the reciprocal vectorsof the 2D lattice formed by the transverse coordinates of thecenters of the chains. We define the dielectric function e(q)of the system as the ratio V ext(q)/V (q) for q in the Brillouinzone of this lattice. Via standard algebraic manipulations inthe reciprocal space we arrive at

e~q!5k14pe2

a2

qx2

q21

Bx~q!qx21Bq

2 , ~A5!

Bx~q!5Cx01

4pe2

ka2 (G0

F2~G!qx

21~q1G!2, ~A6!

B54p2a2 (j J i j . ~A7!

In the limit qx!a21, q!a21 we obtain Eq. ~36! repro-

duced below for convenience,

e~q!5k1qx

2

q2krD

22

qx21aq

2 . ~A8!

Here rD5(Cxk/4pe2)1/2a is the Thomas-Fermi screeningradius, Cx.0 is the effective charge compressibility of thesystem,

Cx5Cx01

4pe2

ka2 (G0

F2~G!G2

;2e2

kln

a

a, ~A9!

and a5B /Cx is the dimensionless anisotropy parameter.Let us now discuss some consequences of Eq. ~A8!.

~i! The dielectric function has the same form as inquasi-1D systems with small rs ~see, e.g., Ref. 48!.

~ii! The screening radius rD is of the order of the inter-chain separation a . Therefore, Brazovskii and Matveeen-035413kos results27 for the soliton energy W and the soliton lengthls remain qualitatively correct63 for high rs provided we userD;a .

~iii! The interaction energy U(r) of two pointlike testcharges separated by a large distance r can be calculated bythe Fourier inversion of

U ~q!54pe2

e~q!q2. ~A10!

For ax21r2 @rD

2 one finds ~cf. Ref. 27!

U~r!.e2

2kuxu expS 2 12rD r2Aax21r2 1Aauxu D .~A11!

The potential U(r) is not exponentially small only within theparaboloid r

2 &AarDuxu @cf. Eq. ~30!#. At the surface ofsuch a paraboloid we have r!Aauxu and U(r) acquires asimpler form quoted in Sec. V,

U~r!.e2

2kuxu expS 2 14Aa r2

rDuxu D . ~A12!~iv! Finally, the effective longitudinal and transverse elas-

tic moduli of the system are given by

Y x5Cx

a2a2 ;

e2

ka2a2 , ~A13!

Y5aY x . ~A14!

APPENDIX B: TUNNELING IN A 3D CRYSTAL WITH ALOW IMPURITY CONCENTRATION

The localization lengths j and jx needed for calculationof the VRH transport are determined by long-distance tun-neling of charge excitations. The problem of tunneling isnontrivial because a broad spectrum of charge excitationsexists. Leaving more detailed investigations for future work,we concentrate on two possible tunneling mechanisms: byelectronlike quasiparticles and by many-body excitations, the2p solitons.

In the quasiparticle mechanism the charge is carried by asingle electron while all other electrons remain unperturbedin their quantum ground states. The rational for examiningthis mechanism is its minimal possible tunneling mass. Theproblem of calculating jx and j reduces to the quantummechanics of a single particle in a fixed external potential.Clearly, the optimal tunneling path should go through theinterstitial positions where the energy barrier is the lowest. Itis convenient then to formulate the problem as a problem ona lattice of such interstitial positions. The relevant variablesare the on-site energies and the hopping matrix elements.The on-site energies are all equal to int;e2/ka . The hop-ping terms for the interchain tunneling, t , are determinedby the band structure in the case of tunneling inside a chemi-cally synthesized materials. In the case of tunneling between-15

M. M. FOGLER, S. TEBER, AND B. I. SHKLOVSKII PHYSICAL REVIEW B 69, 035413 ~2004!distant 1D conductors t}exp(2a /aB). The hopping matrixelement for tunneling along the chain, tx , can be estimatedstraightforwardly, with the result tx5 intexp(2C7Ars), C7;1. In the absence of impurities, the problem is reduced tothe propagation through a periodic lattice. The eigenstates inthat model are labeled by wave vectors k, according to thetight-binding dispersion relation

tb~k!5 int22txcos~kxa!22t@cos~kya!1cos~kza!# .

Below the band edge 5 int22tx24t , the eigenstates areexponentially decaying with distance. The corresponding lo-calization ~decay! lengths can be related to the imaginaryparts of the complex k solution of the equation tb(k)5 . Inthe case of interest, ! int ; t ,tx! int , we obtain

jxq5

a

ln~ int /2tx!;

a

Ars, ~B1!

jq 5

a

ln~ int /2t!, ~B2!

where the superscript q stands for quasiparticle. AlthoughEq. ~B2! was derived for a clean system, it is clear thatimpurity do not affect this result unless present in giganticconcentrations (l;a). Indeed, an individual impurity canmodify the local on-site energy by at most a numerical fac-tor, while j(x) depend on the on-site energy only weakly,logarithmically. In principle, impurity clusters with atypi-cally low on-site energies, resonant with , do exist but aswell known from the analysis of the resonant tunneling prob-lem in random systems, such events are exponentially rareand do not contribute to the bulk localization length in anyappreciable manner.

Let us now turn to the soliton mechanism, we attempt toprofit from the fact that in the bulk, the soliton is the chargeexcitation of the lowest possible energy, so that the energybarrier could perhaps be lower and the tunneling more effec-tive. However, in the case of interchain tunneling, this is notthe case. Indeed, the direct tunneling of a soliton to a differ-ent chain is impossible because the soliton is a compositemany-body excitation. The closest to the interchain solitontunneling that one can imagine is a two-stage process, where,one electron first tunnels to the adjacent chain, and then, onthe second stage, it pushes away other electrons in the regionof length ls to form a soliton. Since the initial energy barrieris still int and the charge spreading only increases the tun-neling action, it is clear that such a contrived process offersno advantage compared to the simple one-stage quasiparticlemechanism. Therefore, j is determined by the latter andcoincides with j

q, leading to Eq. ~8!. Note that for the case

of distant chains where t;exp(2a /aB), the correct limit-ing result j5aB is recovered.

In contrast, for the tunneling along the chain the solitonmechanism is the winner. Consider first the longitudinal tun-neling of a soliton in the absence of impurities. Employingthe usual imaginary-time picture, the action for tunneling035413over a distance x@ls can be estimated as the product of theenergy barrier W and the tunneling time t;x/u . Here u,given by

u5S Cxma

D 1/2; e2k\ 1Ars , ~B3!is the sound velocity.64 The tunneling amplitude is of theorder of exp(2Wx/\u). Using W;e2/kls , we arrive at theestimate of the localization length as follows:

jxs;

lsArs

~ l5!, ~B4!

where the superscript s stands for soliton. Clearly, jxs

@jxq

, so that the soliton mechanism dominates the longitu-dinal tunneling. To account for the dilute impurities, weshould add to the above expression for the action an extraterm (x/l)@\Arsln(ls /a)#. Here the factor (x/l) is the averagenumber of impurities on the tunneling path of length x andthe expression inside the square brackets is the action costfor compactification of the charge e from the length ls tolength a and spreading it back during the tunneling througheach impurity. In this manner, we obtain a corrected expres-sion for jx , which coincides with Eq. ~55!.

APPENDIX C: DIMENSIONAL ENERGY ESTIMATES FORTHE COLLECTIVE PINNING

In this appendix we use the ideas of collective pinning toderive the growth of elastic distortions in a quasi-1D crystalpinned by strong dilute impurities. We also derive the esti-mates of the corresponding gain in the pinning energy den-sity.

We start by reformulating the argument leading to Eq.~33! in the language conventional in the literature devoted tocollective pinning.19 To do so we note that since the energyof a given soliton dipole Es depends on the background elas-tic displacement field u , each impurity exerts a force f52]Es /]u;W/a on the crystal. The long-range variationsof u appear in response to such random forces. Let Du(D)be a characteristic variation of u over a distance D in thetransverse direction and let X be a typical distance overwhich a variation of the same order in the x direction buildsup. Our next step is to estimate the total energy E of a vol-umeV5X3D3D ~relative to the energy of a pristine crystal!.

The energy consists of elastic, Coulomb, and pinningparts,

E5Eel1EC1Epin. ~C1!

In its turn, Eel is the sum of the longitudinal and transverseterms,

Eel;Y xS DuX D2

V1YS DuD D2

V . ~C2!-16

~C5!X and Du can now be found by optimizing E for a fixed D.Not surprisingly, we find that X and D are related by the

lowest-order perturbation theory results. It is generally ex-pected that the growth of Du2 with r slows down beyond theLarkin length65 and that adjustments on larger scales do notlead to any substantial increase in the pinning energy density.

VARIABLE-RANGE HOPPING IN QUASI-ONE- . . . PHYSICAL REVIEW B 69, 035413 ~2004!defining equation ~30! of the paraboloid introduced in Sec.V A ~for rD;a case!,

1 Yu Lu, Solitons and Polarons in Conducting Polymers ~WorldScientific, Singapore, 1988!.

2 A.J. Heeger, S. Kivelson, J.R. Schrieffer, and W.-P. Su, Rev. Mod.Phys. 60, 781 ~1984!.

3 N.F. Mott and E.A. Davis, Electronic Processes in Non-Crystalline Materials 2nd ed. ~Clarendon Press, Oxford, 1979!.

4 B.I. Shklovskii and A.L. Efros, Electronic Properties of DopedSemiconductors ~Springer, New York, 1984!.

5 For review, see M.M. Fogler, in High Magnetic Fields: Applica-tions in Condensed Matter Physics and Spectroscopy, edited byC. Berthier, L.P. Levy, and G. Martinez ~Springer-Verlag, Berlin,2002!; M.M. Fogler, cond-mat/0111001 ~unpublished!.

6 J.M. Tranquada in Neutron Scattering in Layered Copper-OxideSuperconductors, edited by A. Furrer ~Kluwer, Dordrecht,1998!; J.M. Tranquada in Ref. 12.

7 R.G. Mani and K.v. Klitzing, Phys. Rev. B 46, 9877 ~1992!.8 W.A. de Heer, W.S. Basca, A. Chatelain, T. Gerfin, R. Humphrey-

Baker, L. Forro, and D. Ugarte, Science 268, 845 ~1995!; O.Chauvet, L. Forro, L. Zuppiroli, and W.A. De Heer, Synth. Met.86, 2311 ~1997!.

9 F.J. Himpsel, K.N. Altmann, R. Bennewitz, J.N. Crain, A. Kira-kosian, J.-L. Lin, and J.L. McChesney, J. Phys.: Condens. Mat-ter 13, 11 097 ~2001!.

10 For review, see G. Gruner, Density Waves in Solids ~Addison-035413In this case, Eq. ~34! is the final estimate of Epin in the ther-modynamics limit.

Wesley, New York, 1994!; G. Gruner, Rev. Mod. Phys. 60, 1129~1988!.

11 Highly Conducting Quasi-One-Dimensional Organic Crystals,edited by E. Conwell, Semiconductors and Semimetals Vol. 27~Academic Press, San Diego, 1988!.

12 For other representative works on the subject, see also in ECRYS-2002, Proceedings of An International Workshop on ElectronicCrystals, edited by S.A. Brazovskii, N. Kirova, and P. Monceau@J. Phys. IV 12 ~2002!#.

13 L.I. Glazman, I.M. Ruzin, and B.I. Shklovskii, Phys. Rev. B 45,8454 ~1992!.

14 H.J. Schulz, Phys. Rev. Lett. 71, 1864 ~1993!.15 X.G. Wen, Phys. Rev. B 42, 6623 ~1992!; D. Boies, C. Bourbon-

nais, and A.-M.S. Tremblay, Phys. Rev. Lett. 74, 968 ~1995!; E.Arrigoni, Phys. Rev. B 61, 7909 ~2000!.

16 A.O. Gogolin, A.A. Nersesyan, and A.M. Tsvelik, BosonizationApproach to Strongly Correlated Systems ~Cambridge Univer-sity Press, Cambridge, 1999!.

17 T. Giamarchi and H.J. Schulz, Phys. Rev. B 37, 325 ~1988!, andreferences therein.

18 C.L. Kane and M.P.A. Fisher, Phys. Rev. B 46, 15 233 ~1992!.19 G. Blatter, M.V. Feigelman, V.B. Geshkenbein, A.I. Larkin, and

V.M. Vinokur, Rev. Mod. Phys. 66, 1125 ~1994!.20 For VRH transport in a model of noninteracting quasi-1D electron

system, see E.P. Nakhmedov, V.N. Prigodin, and A.N. Sam-The Coulomb energy is of the order of U (qx ,q)r2V , whereU (q)54pe2/kq2 is the Coulomb kernel, qx51/X , q51/D are the characteristic wave vectors involved, r5en]xu;enDu/X is the charge density associated with thelonditudinal compression, and n51/aa

2 is the average elec-tron concentration. Below we show that D!X , so that q@qx , U (q);e2D2/k , and finally,

EC;e2

k

D4

X S Duaa2 D2

. ~C3!

The pinning energy can be estimated as Epin;2Du( j f j ,where f j;W/a is the force exerted on the lattice by j thimpurity. The average number of impurities in the volume Vis Ni5NV and f j have random signs; hence, Epin;2(W/a)DuANi or

Epin;2WS Dua

D S Da

D S Xl D1/2

. ~C4!

Combining Eqs. ~A13! and ~C2!~C4!, we arrive at

E5Wa

S Dua

D 2S aX1 D4Xa2 D 2WS Dua D S DaD S Xl D1/2

.X;D2

aAa. ~C6!

Such a paraboloid is an invariable feature of the elastic re-sponse of the quasi-1D crystal to external forces. What is,surprising, however is that Du2;(ls /l)a2 is small and doesnot depend on D, at odds with Eq. ~32!. The resolution ofthis contradiction comes from a realization that what Du(D)really represents is the elastic distortion due to the adjust-ment of the crystal on a single scale D. In fact, there is ahierarchy of smaller scales D ,D/2,D/4, . . . ,r

min, on which

adjustments are approximately independent. The correct es-timate of Du , Eq. ~32!, is obtained once we sum over allsuch scales, Du2;Ma2ls /l , where M;ln(D/rmin) is thenumber of scales. Thereby, we recover Eq. ~32! and as anadditional benefit, we find the expression for the energy ofthe collective pinning,

E;2WS Dua

D 2S Da

D 2. ~C7!Let us define the pinning energy density by Epin5E/XD2.Using Eqs. ~28!, ~33!, ~C6!, and ~C7! we obtain the estimateof Epin at the Larkin scale as given by Eq. ~34!. As clear fromthis derivation, both Eqs. ~32! and ~34! are essentially the-17

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