PHYSICAL REVIEW B 88, 075139 (2013)

Disordered conduction in single-crystalline dimer Mott compounds

Majed Abdel-Jawad,1,* Naoya Tajima,1 Reizo Kato,1 and Ichiro Terasaki21RIKEN, Hirosawa, Wako, Saitama 351-0198, Japan

2Department of Physics, Nagoya University, Furocho, Chikusa, Nagoya 464-8602, Japan(Received 25 November 2012; revised manuscript received 30 July 2013; published 23 August 2013)

We study the temperature and frequency dependence of the dielectric constant and the ac conductivity in thedimer Mott insulator family β ′-type Pd(dmit)2 (where dmit denotes 1,3-dithiole–2-thione–4,5-dithiolate). Wefind that all compounds within this family have relaxorlike dielectric relaxation with a Curie-Weiss increasein the dielectric constant with decreasing temperature. An analysis of the dielectric data shows a clear relationbetween electronic correlation and the dipole-dipole interaction; also, geometrical frustration appears to weakenthis interaction. Furthermore, the dc and ac conductivities are shown to be closely correlated and follow the sameuniversal ac conductivity as disordered solids.

DOI: 10.1103/PhysRevB.88.075139 PACS number(s): 77.84.Jd, 72.80.Le, 72.80.Ng, 77.22.Gm

I. INTRODUCTION

The electrical properties of glassy states have been exten-sively studied. Materials as diverse as amorphous semiconduc-tors, ionic and electronic conducting polymers, ion conductingglasses, polycrystalline semiconductors, and doped single-crystal semiconductors at low temperatures have very commonelectrical properties.1 For instance, the ac conductivity σ (ω) ofthese materials can be scaled into a single curve independentof temperature and their dielectric constants ε have relaxationssimilar to dipolar liquids.1 The relaxations in question are notindependent from the dc conductivity σ (0), but are related bythe Barton-Nakajima-Namikawa (BNN) relation

σ (0) = pε0�εωm, (1)

where �ε = ε(0) − ε(∞) is the dielectric loss strength, p isa numerical constant close to 1, and ωm is the dielectric losspeak angular frequency. The universality of those properties issuch that we cannot distinguish the electrical response betweenelectronic and ionic conductors.1

Molecular conductors are generally thought to form veryclean single crystals. For insulating materials at ambientpressure, an application of pressure generally induces atransition toward a metallic state. The typical mean free pathof the charge is in the neighborhood of thousands of angstromsat the lowest temperatures.2 As such, the electronic nature ofthe insulating states of those compounds is considered to bevery homogeneous, as only a very low level of crystallographicdefects results in such high values of the mean free path.

In molecular-based dimer Mott insulators, the conductingmolecules are arranged such that molecule pairs with a largetransfer integral can be defined as a unit. The band fillingis in an effectively half-filled state. Geometrically frustratedarrangements of those units, e.g., a triangular lattice,3 cansuppress the ground-state magnetic ordering. However, mostcases are of the highly correlated and weak frustration type,with an antiferromagnetic (AFM) ground state expected froma Mott insulator.

Most of the theoretical works on dimer Mott insulatorshave been based on the half-filled single-band Hubbard model,namely, the dimer Mott model. Although this model neglectsall charge degrees of freedom within the dimer unit, it correctlydescribes at least qualitatively the phase diagram of the dimer

Mott systems under pressure. A few theoretical investigationshave questioned the use of this model on various grounds,such as a very weak degree of dimerization4 or the absenceof superconductivity.5 Our dielectric constant measurementsin κ(BEDT-TTF)2Cu2(CN)3 [κ-(CN)3],6 where BEDT-TTFstands for Bis(ethylenedithio)tetrathiafulvalene, revealed thatthe temperature and frequency dependence of ε was similar tothat of a relaxor ferroelectric compound. This result, combinedwith the 6-K anomaly found in thermal expansion,7 specificheat,8 and thermal conductivity9 data, has encouraged10 andinspired11,12 subsequent theoretical work focusing on thecharge degree of freedom within the dimer unit. It has beenpreviously noted that similar relaxor behavior for the dielectricconstant was also present in a quasi-one-dimensional dimerMott insulator β ′-(BEDT-TTF)2ICl2; furthermore, this effectwas probably common to the whole BEDT-TTF-based dimerMott insulators.6

In this paper we report on the frequency and temperaturedependence of ε within the dimer Mott insulator β ′-Pd(dmit)2

(where dmit denotes 1,3-dithiol–2-thione–4,5-dithiolate) salts.Compared to BEDT-TTF molecules, the Pd(dmit)2 moleculehas a similar planar π -conjugated structure with two no-table differences: The central carbon-carbon double bondis replaced by a Pd metal ion, and there are no terminalethylene groups.13 The absence of terminal ethylene groupsis important within this study, as they are a well-known sourceof disorder within the BEDT-TTF compounds that can affectthe electronic properties.14 Similar to the κ-type BEDT-TTFsalts, the β ′-type of Pd(dmit)2 salts are generally describedin terms of dimer units of Pd(dmit)2 that are arranged in aquasitriangular lattice;15 however, the dimer units are stackedin a face-to-face configuration with a degree of electronicdimerization greater than that of the κ-BEDT-TTF salts.15 Atambient pressure, three different ground states can be found inβ ′-(cation)[Pd(dmit)2]2: cation denotes Me4P, Me4As, Me4Sb,EtMe3As, Et2Me2As, and Et2Me2P for the AFM insulators(Et denotes C2H5- and Me denotes CH3-); cation denotesEtMe3Sb for the spin liquid (SL) candidate; and cationdenotes Et2Me2Sb for the diamagnetic charge-ordered (CO)insulator.15,16 A detailed study of the CO state is beyond thescope of the present paper.

We find that all β ′-Pd(dmit)2 salts have a relaxorlikedielectric constant with a different temperature scale unique

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ABDEL-JAWAD, TAJIMA, KATO, AND TERASAKI PHYSICAL REVIEW B 88, 075139 (2013)

to each cation. The electronic correlation and the degree ofgeometrical frustration are found to affect the dipole-dipoleinteractions, while the dipole moment seems to be independentof those two parameters. We emphasize in this paper that theconductivity and dielectric constant of dimer Mott insulatorsfollow the same universal scaling and relationship as adisordered solid, while the crystal structure is at the clean limit.

II. EXPERIMENT

Single crystals of β ′-(cation)[Pd(dmit)2]2 were prepared bythe air oxidation of (cation)2-Pd(dmit)2 in an acetone solutioncontaining acetic acid at 5 ◦C–10 ◦C. Dielectric constant andac conductivity measurements were conducted out with anautobalancing bridge (Agilent E4980A impedance analyzer).The samples were mounted in a parallel-plate setup andwere cooled to a cryogenic temperature within a variabletemperature insert placed into a liquid helium cryostat. Themeasured directions of all the samples presented here werelimited to the out-of-plane direction of the crystal, the c

axis, as the parallel-plate setup required thin samples withlarge surfaces that are produced by crystal growth onlyalong the out-of plane direction. Gold paste or carbon pastewas applied on opposite sides of the crystal surface as theparallel-plate electrodes. Both conducting pastes gave similarresults. Furthermore, to ensure consistency of our results, atleast four samples of each cation were measured from differentcrystal batch growths. Similar to the κ-(CN)3 case,6 we notethat the values of ε′

r are independent of dc bias voltage upto the dielectric breakdown levels, which is in contrast tothe relaxors.17 We further note the absence of any detectablehysteresis in ε′

r, which excludes the extrinsic18 and internalbarrier layer19 as the origin of the dielectric response. Thebandwidth W and the intradimer transfer integral tA valueswere calculated with a dimer Mott tight-binding model based

on the extended Huckel method.20 Then tA/W represents theelectron correlation, as the on-site Coulomb repulsion U isapproximately proportional to tA.21

III. RESULTS AND DISCUSSION

Figure 1 shows the typical behavior of the real part of εr

measured in various β ′-Pd(dmit)2 compounds. The generaltrend is similar and follows the same response seen in relaxorferroelectrics22 and κ-(CN)3,6 with strong frequency disper-sion and an increase of ε′

r with decreasing temperature. Thisfollows a Curie-Weiss law (Fig. 3, bottom inset), defined as

�ε(T ) = C

T − TCurie, (2)

where C is the Curie constant and TCurie is the Curietemperature. Figure 2 (top inset) shows the typical behaviorof the frequency dependence of ε′

r relaxation in β ′-Pd(dmit)2.This relaxation is best fitted by a Fourier-transformed stretchedexponential function, the Kohlrausch-Williams-Watts (KWW)function typically used in glassy states. The dielectric functionis then defined as

ε(ω) = ε∞ + (εs − ε∞)∫ +∞

0−de−(t/τ )β

dte−iωtdt, (3)

where τ is the relaxation time and β is the stretching exponent.Equation (3) reduces to a simple Debye relaxation when β = 1.If the entire relaxation is not within our frequency windowof 20 Hz to 1 MHz, the KWW model is overparametrized andwe cannot estimate the parameters of this model reliably. Thetemperature range where all the relaxation occurs within ourfrequency window varies between compounds. In the case ofEtMe3Sb[Pd(dmit)2]2, for instance, this temperature range isfrom roughly 30 to 40 K. Reliable estimates of the stretching

FIG. 1. Frequency and temperature dependence of the real part of the relative dielectric constant of (cation) [Pd(dmit)2]2, where thefrequency range and the cation are specified within the graph. The inset shows the ac conductivity of the respective compounds.

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FIG. 2. (Color online) Taylor-Isard scaling of the ac con-ductivity data of Me4P[Pd(dmit)2]2, Me4Sb[Pd(dmit)2]2, andEtMe3Sb[Pd(dmit)2]2 with frequencies from 500 Hz to 50 kHz with31, 5, and 80 steps, respectively, with log10 scaling. The top insetshows the frequency dependence at 35 K of the same data set withfrequencies from 177 Hz to 1 MHz with 150 steps spread out on alog10 scale and Debye and KWW fits with β = 0.49 at the dielectriclevel. The bottom inset shows a plot of the 21.1-Hz conductivity andε0�εωm.

exponent value β are found to never exceed the value of 3/5 andto decrease with decreasing temperature in all β ′-Pd(dmit)2.

The mean relaxation time 〈τ 〉 can be deduced from theKWW fits of the frequency dependence or by estimating themidpoint of the relaxation when ε′(〈τ 〉)(T ) = ε′(+∞)(T ) +δε(T )/2 is true. Although both methods yield similar results,the midpoint of the relaxation method was used as thetemperature range where we can estimate 〈τ 〉 reliably is greaterthan that from the KWW method. Within this temperaturerange, we find that 〈τ 〉 follows a simple Arrhenius temperaturedependence with an activation gap Ea similar to that estimatedfrom the dc conductivity. The BNN relation is obeyed in allβ ′-Pd(dmit)2 salts, as shown, for instance, in the SL sample inFig. 2 (bottom inset). The implication of the BNN relation isthat the dc conductivity and the excess polarization share thesame physical origin. Then �ε may arise entirely from mobilecharges as from a dipolar polarization perspective and thehopping of a charge between neighboring sites can be seen asthe rotation of a permanent dipole.23 However, this is unlikelyas the estimated hopping distance or the mean displacement23

are less than 1 A, while the minimum distance between the siteswithin the dimer unit is longer than 3 A. Moreover, �ε followsa Curie-Weiss temperature dependence, while models basedpurely on hopping conduction yield a simple 1/T function.The frequency dependence of the dielectric response is also ofthe dipolar-dominated response type1 and the ferroelectricityhas been recently reported in κ-(BEDT-TTF)2Cu[N(CN)2]Cl(Ref. 24) and β ′-(BEDT-TTF)2ICl2.25 As such, we still believethat bulk dielectric dipoles6 are the main contributors to �ε.

Dipolar disorder is found to significantly affect the chargemobility in molecularly doped polymers26,27 in which bothcharge carriers and permanent dipoles are present. As stated

previously, charge-transfer salt single crystals are of such highquality that the disorder cannot have a crystallographic nature.However, Fig. 2 shows that the ac conductivity can be scaledinto a master curve whose shape is similar between cations andthose found in ion conducting glass, conducting polymers,and other highly disordered compounds.1 The flat regionof this Taylor-Isard scaling consist of the high-temperaturefrequency-independent part of the ac conductivity, while thecurved region consist of the frequency-dependent part of theac conductivity. Its significance lies in the shape of the mastercurve being independent of the details of the system eventhough it approximates �ε as a simple 1/T Curie dependence.This universal behavior of the ac conductivity to the details ofthe disorder occurs when the local mobility covers many orderof magnitude and approaches the extreme-disorder limit.1 Thehopping of charge carriers in a random energy landscape, asin the symmetric random-barrier model,1 describes well allthe major features of the conductivity. It is then possible thatthis random energy landscape originates from the randomlyoriented polar nanoregions,27 which give rise to the relaxorlikedielectric constant. Percolation then explains the commonEa between σ (0) and 〈τ 〉 of the polar nanoregions. Thelargest energy barriers between the percolation clusters act asbottlenecks for σ (0) (Ref. 1) and delimit the polar nanoregions.

Since �ε follows a Curie-Weiss temperature dependence,TCurie should reflect the dipole-dipole interaction strength.Plots of TCurie as a function of the electronic on-site correlationare shown in Fig. 3 (top). A linear increase of the Curietemperature with increasing electronic correlation is clearlyvisible for β ′-Pd(dmit)2 compounds with low geometricalfrustration. This increase of the dipole-dipole interaction withincreasing electronic correlation is to be expected if the dipolenature is electronic. However, above some critical value, TCurie

appear to decrease with further increases of tA/U . We attributethis decrease of TCurie to geometrical frustration as the increaseof tA/U is small in comparison to the increase of the degreeof frustration t/t ′. In addition to this, the decrease of TCurie

coincides with values of t/t ′ in proximity to 1, where aspin-liquid state is thought to exist (see Fig. 3, top inset). Thisrelation with t/t ′ implies that spin interactions are assistingthe electronic ordering as predicted in the paired-electroncrystal model.10 A comparison of the Neel temperature TN andTCurie shows the interesting case in which an AFM orderingtemperature is greater than the dipole ordering in Me4P (TN ≈40 K) and Me4As (TN ≈ 35 K). Previous reports of TCurie

in κ-BEDT-TTF salts6,24 and β ′-(BEDT-TTF)2ICl2 (Ref. 25)were above or equal to TN . Those results gave the impressionthat the charge freezing was essential for stabilizing the AFMstate. Our result seems to contradict this hypothesis at first, butwe should note that the AFM transition in Me4P[Pd(dmit)2]2

is unusually weak.29

As in Ref. 6, we converted the Curie constant estimatesinto the charge disproportionation values shown in Fig. 3(bottom). Illustrations of what the charge disproportionationpattern may look like can be found in Ref. 25, as β ′-Pd(dmit)2

salts have a similar arrangement of the conducting moleculesto β ′-(BEDT-TTF)2ICl2. Surprisingly, the estimated chargedisproportionation appears to have only small changes withinthe β ′-Pd(dmit)2 salts, implying that the dipole momentsare independent of U/W and spin fluctuations. We note,

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ABDEL-JAWAD, TAJIMA, KATO, AND TERASAKI PHYSICAL REVIEW B 88, 075139 (2013)

FIG. 3. (Color online) The top is a plot of the Curie temperaturefrom the dielectric constant as a function of on-site correlation.20 Thetop inset shows the relation between the electronic correlation and thefrustration parameter t ′/t in the β ′-Pd(dmit)2 salts.20 The bottom isan estimate of the charge disproportionation as a function of on-sitecorrelation from the Curie constant, where 0 indicates equal chargeoccupancy of the two molecules of a dimer structure. The lower(blue) dashed line is related to the minimum dipole moment value fora three-dimensional state converted to charge disproportionation.28

The bottom inset shows a plot of the inverse dielectric constantin EtMe3Sb[Pd(dmit)2]2 used to estimate the Curie parameters (reddashed straight line).

however, that changes in tA/W reflect mostly those in Wbecause the distance between molecules forming the dimerunit does not change significantly in the β ′-Pd(dmit)2 salts.20

The distance between those molecules determine tA, fromwhich U and the degree of dimerization is estimated; thus

we can infer that the dipole moment is probably related tothe degree of dimerization. Although further measurementsare required to confirm this hypothesis, we highlight thatκ-(CN)3, which has a lower degree of dimerization30 thanβ ′-Pd(dmit)2 (Ref. 20) salts, has a lower value of the estimatedcharge disproportionation at 0.1 (Ref. 6) and 0.13.25 Froma theoretical viewpoint, the paired-electron crystal model10

cannot explain the trend of the dipole moment; however,the quantum dipole moment11 model can explain it if thenearest-neighbor Coulomb repulsion V is also independentof the cation in β ′-Pd(dmit)2 salts.

IV. SUMMARY

In conclusion, our dielectric and conductivity data showa direct link between dc and ac conductivity through theBNN relation and confirm that both are electronic in origin.This electronic origin of the excess polarization is furtherconfirmed with the correlation between the Curie temperaturedependence and the electronic properties of β ′-Pd(dmit)2 salts.In this case, the dipole-dipole interaction is found to depend onboth the electronic correlation and the spin interaction, whilethe dipole moment is found to be independent of interdimertransfers integrals. Most importantly, the BNN relation, theKWW frequency dependence of the relaxation, and the acconductivity scaling into a single master curve all confirm thatthe conductivity in dimer Mott compounds is electronicallydisordered,1 but in a crystallographically clean system. Oneof the implications of this result is that the metal-insulatortransition occurs between a clean correlated metal and an elec-tronically correlated and disordered insulator. This insight intothe electronic nature of dimer Mott compounds should helpus understand some of the peculiar properties of dimer Mottinsulators, such as the increase in conductivity with additionalcrystallographic disorder31 in the insulating state, whereasthese same types of defects decrease the conductivity in themetallic state.32 The nature of disorder at the electronic levelwithin dimer Mott insulators has been reported previously;33

here we state that this behavior is intrinsic to all dimer Mottsystems.

ACKNOWLEDGMENTS

The authors would like to thank Takahiko Sasaki forhelpful discussions. This research was partially supported by agrant-in-aid for Scientific Research on Innovative Areas (GrantNo. 20110003) from the Ministry of Education, Culture,Sports, Science and Technology.

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