13 C NMR Study on the Charge-Disproportionated Conducting State in the Quasi-Two-Dimensional Organic Conductor α -(BEDT-TTF) 2 I 3
Michihiro Hirata, Kyohei Ishikawa, Kazuya Miyagawa, Kazushi Kanoda
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t C NMR Study on the Charge-Disproportionated Conducting State in theQuasi-Two-Dimensional Organic Conductor α -(BEDT-TTF) I Michihiro Hirata, Kyohei Ishikawa, Kazuya Miyagawa and Kazushi Kanoda
Department of Applied Physics, University of Tokyo, Bunkyo City, Tokyo, 113-8656, Japan
Masafumi Tamura
Department of Physics, Faculty of Science and Technology,Tokyo University of Science, Noda, Chiba, 278-8510, Japan (Dated: August 3, 2018)The conducting state of the quasi-two-dimensional organic conductor, α -(BEDT-TTF) I , at am-bient pressure is investigated with C NMR measurements, which separate the local electronicstates at three nonequivalent molecular sites (A, B, and C). The spin susceptibility and electroncorrelation effect are revealed in a locally resolved manner. While there is no remarkable site-dependence around room temperature, the local spin susceptibility gradually disproportionatesamong the nonequivalent sites with decreasing temperature. The disproportionation-ratio yields5:4:6 for A:B:C molecules at 140 K. Distinct site- and temperature-dependences are also observedin the Korringa ratio, K i ∝ (1 /T T ) i K − i ( i = A, B, and C), which is a measure of the strength andthe type of electron correlations. The values of K i point to sizable antiferromagnetic spin correla-tion. We argue the present results in terms of the theoretical prediction of the peculiar site-specificreciprocal-space ( k -space) anisotropy on the tilted Dirac cone, and discuss the k -dependent profilesof the spin susceptibility and electron correlation on the cone. PACS numbers: 76.60.-k, 71.20.-b, 71.30.+h
I. INTRODUCTION
The organic charge-transfer salt α -(BEDT-TTF) I (abbreviated as α -I hereafter) is a quasi-two-dimensional (Q2D) electron system with a 1/4-filledhole band which exhibits strong electron correlationeffects [BEDT-TTF (ET hereafter) is the abbreviation ofbisethylenedithio-tetrathiafulvalene; see Fig. 1(b)]. α -I is comprised of alternately stacked 2D conducting layersof (ET) +1 / molecules and non-magnetic insulating lay-ers of triiodide anions (I ) − [Fig. 1(a)]. The unit cellcontains four ET molecules, three of which are crystal-lographically nonequivalent [i.e., A, B, and C molecules;see Fig. 1(c)]. At room temperature, the band structurecalculation predicts semi-metallic Fermi surfaces. Withdecreasing temperature, resistivity shows weak temper-ature dependence down to T CO ∼
135 K, at which thesystem undergoes a first-order phase transition froma paramagnetic conductor to a nonmagnetic insulatoraccompanied by an inversion symmetry breaking. Below T CO , electrons are localized on ET lattice sites,forming the horizontal strip type of charge ordering(CO), which is believed to be stabilized by thelong-range Coulomb interactions. Under hydrostatic pressures, the phase transitiontends to be suppressed.
Above P C ∼ . revealed that an unusualelectronic structure with the linear dispersion is realizedaround the Fermi level, ε F , (i.e., massless Dirac cone)which is partly confirmed by recent magnetotransport experiments. In contrast to the graphene monolayer,known as the typical massless Dirac-cone system withisotropic dispersion, the Dirac cone in α -I is stronglytilted; the conical slant varies 10 times around the apexin the 2D k -space. The tilt stems from the low lat-tice symmetry in α -I possessing only inversion symme-try (space group P -1). Because of this anisotropy of theband dispersion, a large imbalance in the local electronicdensity of states around ε F is predicted among inde-pendent A, B, and C molecules. Recent C nuclearmagnetic resonance ( C NMR) experiments under hy-drostatic pressures observed large difference in the localspin susceptibilities at these molecular sites.
The Dirac-cone picture has been discussed so far onlyat high hydrostatic pressures (
P > P C ) in α -I . How-ever, several experiments have recently suggested thatit might be also applicable to the low-pressure conduct-ing phase. Resistivity, Raman,
10 13
C NMR, x-ray, Hall, and thermopower measurements in-dicate that the ambient- and high-pressure conductingphases have many properties in common. Moreover, theband structure calculation suggests that the conicalapex (Dirac point) locates at ∼ ε F even atambient pressure, and can be pushed up to ε F by in-troducing a site-dependent potential imbalance betweenA, B, and C nonequivalent molecules. Aside fromthis, the ambient-pressure conducting phase exhibits sev-eral anomalous properties such as non-Drude optical con-ductivity just above T CO , and a large charge densitydisproportionation among A, B, and C molecular sitesas observed in x-ray, Raman, and C NMR exper-iments. On the whole, these findings probably indicatethat the conducting state in α -I is closely associatedwith the Dirac cone and/or CO fluctuations. However,the local electronic properties are not yet fully under-stood quantitatively, in particular, the electron correla-tion effects.The purpose of the present study is to clarify the elec-tronic nature in the ambient-pressure conducting state of α -I in microscopic and site-specific manners. Utilizing C NMR experiments, we explored the local spin suscep-tibility and electron correlation effects at each nonequiv-alent molecule ( i = A, B, and C), since NMR is capableof probing the local electronic states at these indepen-dent molecules separately. As the hyperfine coupling ofa nuclear spin with its surrounding electrons is highlyanisotropic at the C site, one should know the field-angulear-dependent characteristics of NMR spectra tomake a reliable assignment of the NMR line and deter-mine hyperfine tensors, which correspondingly providesa quantitative basis for deducing the local spin suscep-tibility and the electron correlation effects. Thus, wemeasured the angular dependences of the NMR Knightshift, K i , with rotating the external field in the planesparallel and perpendicular to the conducting layers [i.e.,the ab plane in Fig. 1(c)] at each temperature measured.The C hyperfine-shift tensors and the local spin sus-ceptibility were determined from the angular-dependentprofile of K i at each independent molecule. The nuclearspin-lattice relaxation rate, (1 /T ) i , was measured un-der the in-plane fields in order to evaluate the Korringaratio, K i ∝ (1 /T T ) i K − i , which represents the strengthand type of the electron correlations. The results are dis-cussed in connection with the effect of CO fluctuationsand Dirac cone. II. METHODS
To evaluate the local electronic properties in theambient-pressure paramagnetic conducting state of α -I ,we performed C NMR measurements at external field H of 6.00 T applied in the ab and bc planes in the tem-perature range from room temperature down to 140 K( > T CO ). We grew the single crystal of α -I of the size2 . × . × . by the conventional electrochemicaloxidization method. The central double-bonded carbonatoms in ET were selectively enriched with C isotopes(nuclear spin I = 1 / γ n / π = 10.705 MHz/T) at 99%concentration [see Fig. 1(b)], which means that the ob-served C NMR spectra come from the central carbonsites. The C NMR spectra were obtained by the fastFourier transformation of the spin echo signals. As theorigin of the NMR shift, we referred to the resonancefrequency of the C NMR signal of TMS [tetramethyl-silane, (CH ) Si]. The spin-lattice relaxation rate, 1 /T ,was obtained by the standard saturation and recoverymethod. III. RESULTSA. Angular dependence of the C NMR lineshapes and line assignment
The assignment of the observed C lines in the NMRspectra to A, B, and C nonequivalent molecular sites isthe starting point of the present work. This can be ac-complished by considering the crystal structure, the sym-metry of A, B, and C molecules in the crystal, and thenuclear dipolar splitting in the analysis of the angulardependence of C NMR spectra.In ET-based organic conductors, all crystallographi-cally nonequivalent molecules contribute to distinct Clines in the C NMR line shapes. In the conducting stateof α -I , at least three peaks are thus expected associatedwith the nonequivalent A, B, and C molecules [Fig. 1(c)].However, each molecule has two neighboring C nuclei[Fig. 1(b)], which are coupled by the nuclear dipole inter-action. This coupling makes a single line split into a dou-blet or a quartet depending on whether the molecule hasinversion symmetry or not, respectively.
The Pakedoublet structure should be observed when the neighbor-ing nuclei are equivalent, which is the case in molecules Band C. The Pake doublet is characterized by the nucleardipolar splitting width (kHz), d , given by d = 32 γ ~ r (1 − ζ i cos η i ) , (1)where γ n is the gyromagnetic ratio of C nucleus, 2 π ~ is Planck’s constant, r = 1.360 ˚A is the distance be-tween two C nuclei, and ( ζ i , η i ) are the angles formedbetween the direction of H and the C= C vector ofthe i -site molecule ( i = A, B, and C), as illustrated inFig. 3(b). The quartet structure is expected when theneighboring C sites are nonequivalent as in moleculesA. The quartet is composed of four lines ( ν = 1 − the line shift δ ν of which is dependent on the nucleardipolar splitting width d and the NMR shift differencebetween the two neighboring C sites, ∆ δ ; i.e., δ ν = δ ± d/ H ± p ( d/ H ) + (∆ δ ) /
2, where δ is the aver-aged shift of the quartet, and equals to the average of theNMR shift at the neighboring C sites in ET without thenuclear dipole coupling. Thus, in the present case, eight C lines composed of two Pake doublets and one quartetare expected in the C NMR spectra.In order to assign the NMR lines properly, we measuredthe magnetic-field angular dependences of the NMR spec-tra. Figure 2 presents the typical data at 260 K with theexternal field H applied within the ab plane ( H//ab ) and bc plane ( H//bc ). The line shapes are strongly depen-dent on the direction of H . In Fig. 2(a), the C linescan be divided into two groups, exhibiting an anti-phaseangular dependence as the direction of H , ψ , is changedwithin the ab plane [ ψ = 0 ◦ is set as H//a ; see the in-set of Fig. 2(a)]. In Fig. 2(b), on the other hand, all the C lines exhibit an in-phase variation when H is rotatedin the bc plane [ θ = 0 ◦ is set as H//c ; see the inset ofFig. 2(b)]. These phase relations can be adequately un-derstood from the crystal structure; i.e., the directionof C-2 p z orbitals within the unit cell. It is well knownthat the 2 p z electrons of C atoms produce a dipole-type hyperfine field at the C nuclear position, and givethe largest contribution to the C NMR shift.
Asschematically shown in Fig. 1(c), molecules B and C, andmolecules A form respectively a stack along the a axis.These stacks build a fishbone-like molecular arrangementin the ab plane, and are approximately connected withmirror operation [( a , b , c ) → ( a , − b , c )]. The 2 p z orbitals,pointing perpendicular to the molecular plane [along the z axis in Fig. 3(a)], are parallel within the same stack,while they are mutually perpendicular between differentstacks. This geometry gives rise to an anti-phase relationin the angular dependence of the NMR shift when the ex-ternal field is rotated in the ab plane. On the other hand,all the 2 p z orbitals are roughly confined to the ab plane,which is expected to cause an in-phase dependence of theNMR lines when the filed is rotated within the bc plane.Referring to the considerations given above, we first as-signed the C lines in Fig. 2(a). The four lines denotedby open circles are attributable to the quartet (moleculesA), and the rest four lines to two pairs of the Pake dou-blets (molecules B and C), as shown in Fig. 2(a). Itis easy to distinguish molecules B and C when one fo-cuses on the dihedral angles between molecules A and B( ∼ ◦ ), and A and C ( ∼ ◦ ). Since the former islarger than the latter, molecules B should exhibit largerphase difference from molecules A than molecules C willin the angular dependence of the line shapes. We thusassigned the C lines denoted by triangles and crossesto molecules B and C, respectively, as shown in Fig. 2(a).The results are consistent with the previous work. InFig. 2(c), the observed angular dependences of the nu-clear dipolar splitting width d for A, B, and C moleculesare presented with the corresponding symbols. The solid,dotted, and dashed lines are calculated angular depen-dences of d for A, B, and C molecules, respectively, with r = 1.360 ˚A and the molecular orientations obtained fromRef. 34. The data agree well with the calculations.Next, we focus on Fig. 2(b). By comparing the NMRspectra at θ = ψ = 90 ◦ in Figs. 2(a) and 2(b) (i.e., H//b ), the C lines under
H//bc were assigned as shownin Fig. 2(b) [with the same symbols as in Fig. 2(a)]. Fig-ure 2(d) presents the angular dependences of the dipolarsplitting width d under H//bc . Calculated angular de-pendences are presented together, which properly repro-duce the experimental results. All of these analyses thusassure our assignments comprehensively.
B. Angular dependence of the C NMR shift anddetermination of the hyperfine-shift tensors
The C NMR shift is proportional to the hyperfinefield at the C site projected onto the direction of theexternal field H . The C NMR shift at molecule i (= A, B, and C), δ i , is defined as the averaged shift of the linesat molecule i . In Fig. 4, we present the representativeangular dependences of the C NMR shift δ i at 260 Kunder H//ab [Fig. 4(a)] and
H//bc [Fig. 4(b)]. δ i is ex-pressed as δ i = [ δ xxi ( H x ) + δ yyi ( H y ) + δ zzi ( H z ) ] / | H | ,where x , y , and z are the ET principal axes [Fig. 3(a)], H = ( H x , H y , H z ) is the external field in the ET princi-pal axis [Fig. 3(b)], and δ xxi , δ yyi , and δ zzi are the principalcomponents of the C-hyperfine-shift tensor. [Note thatthe shift tensor for the molecule A with two nonequiva-lent C sites is defined as their average, although it canbe determined site specifically. The reason why we do sois that the charge and spin densities discussed below interms of the shift is molecular-specific, and we need theshift value representative of the molecular site. Moreover,this definition is beneficial for the evaluation of electroncorrelation effects described in Section III D).] We fit theexpression to the data in Figs. 4(a) and 4(b), using thecrystal structure data given by Sawa et al . The param-eters optimized are listed in Table I. The fits in Fig. 4properly capture the sinusoidal angular dependences of δ i under both field geometries. As seen in Table I, theshift tensors have different principal values among A, B,and C sites. This evidences differentiations in local spinsusceptibility and charge density among these nonequiv-alent sites in the unit cell. C. Temperature dependence of the C NMR shiftand the local spin densities
Next, we proceed to separate the local charge den-sity and spin density contributions in the observed shiftdata, and investigate the disproportionation of the lo-cal susceptibility within the unit cell and its anoma-lous temperature dependence. Figures 5(a) and 5(b)show the temperature dependences of the C NMRspectra at ψ = 110 ◦ and 50 ◦ (under H//ab ), respec-tively. Large temperature dependences are seen in theNMR shift δ i when | δ i | is large. On the other hand, δ i is almost temperature independent when | δ i | becomessmall. The shift δ i is the sum of the Knight shift andthe chemical shift ( δ i = K i + σ i ), where the Knightshift, K i , is related to the local spin susceptibility χ i through K i = a i χ i and the chemical shift, σ i , origi-nates from the orbital motion of electrons within ETmolecules. a i is the hyperfine coupling constant be-tween the C nuclear and itinerant carrier spins. It iswell known that the C chemical shift in ET dependson the molecular valence; namely, the amount of hole onET. Because large charge density disproportionationshave been observed in α -I among nonequivalent A, B,and C molecules above T CO , σ i should be de-termined for each molecule in this compound. Using theprincipal values of the chemical-shift tensors σ xxi , σ yyi ,and σ zzi (in ppm) determined at 60 K, and linearlyinterpolating the molecular valence, ρ i , determined byX-ray diffraction to estimate ρ i at 60 K, we get the fol-lowing relations between the chemical-shift tensors andvalence ρ i ; σ xxi = 130 . ρ i + 46 . σ yyi = 30 . ρ i + 159 . σ zzi = − . ρ i + 61 .
9. In our case, the temperaturedependence of σ i (in ppm) is negligibly small, and thevalue of σ i is determined as σ A ∼ σ B ∼ σ C ∼
93 at ψ = 110 ◦ , and σ A ∼ σ B ∼ σ C ∼
166 at ψ = 50 ◦ , respectively. In general, thesechemical-shift tensors allow us to calculate the chemicalshift σ i under arbitrary field orientation. Thus, to de-termine the angular dependence of the Knight shift inthe following, we calculated the angular dependences of σ i site-selectively and subtracted them from those of theshift δ i .Figures 5(c) and 5(d) depict the temperature depen-dences of the NMR Knight shift, K i (= δ i − σ i ), under ψ =110 ◦ and 50 ◦ ( H//ab ), respectively, with the use of thechemical shifts σ i determined above. The Knight shift K i exhibits strong anisotropy within the ab plane; for in-stance, K A is almost zero in Fig. 5(c), while it is largein Fig. 5(d). Since the total spin susceptibility, χ spin ( ∝ χ A + χ B + χ C ), is isotropic in α -I , the anisotropic be-havior of K i should stem from the anisotropy of the hy-perfine interaction at the C position, which is expressedas K i = [ a xxi ( H x ) + a yyi ( H y ) + a zzi ( H z ) ] χ i / | H | with a xxi , a yyi , and a zzi the principal components of the Chyperfine-coupling tensor [see Fig. 3(a)]. [This meansthat the small value of K A in Fig. 5(c) should be at-tributed to the vanishingly small hyperfine-coupling con-stant at ψ = 110 ◦ geometry.] In ET-based materials, it isknown that the profile of the highest occupied molecularorbital (HOMO) of ET determines the anisotropy of thecoupling tensor a µµi ( µ = x , y , and z ). The temper-ature dependence of a µµi is negligible, since the spatialdistribution of HOMO is most likely to be temperatureindependent. Moreover, the spatial profile of HOMOis reasonably assumed to be the same at all moleculesin the unit cell because the observed differences in themolecular structures are very small. Therefore, in thefirst approximation, we can assume that a µµi is site- andtemperature-independent. We also note that moleculesA, B, and C are arranged in a nearly symmetrical mannerwithin the ab plane, as we mentioned above [Fig. 1(c)].Hence, if H is rotated in the ab plane, the average, K ave i ,and the amplitude, K amp i , of the angular dependences ofthe Knight shift [see Fig. 6] are both expected to reflectthe magnitude of χ i ; K ave i = a ave χ i and K amp i = a amp χ i with a ave and a amp the averaged hyperfine-coupling con-stants.Figure 7 shows the temperature dependences of K ave i and K amp i . For determining them, we calculated the an-gular dependence of σ i , using the empirical relations de-duced above, and subtracted them from the angular de-pendences of the shift, δ i , at all temperatures measured.The external field H was rotated in the ab plane, because K amp i becomes largest in this field geometry and the lineshapes are readily discerned owing to the anti-phase an-gular dependence of the lines as mentioned above (inSec. III A). Around room temperature, there are lit- tle site-dependences in the observed K ave i and K amp i .With temperature decreased, however, they exhibit site-specific temperature dependences, and then begin to de-crease monotonically below T ∼
180 K at all sites downto T CO ∼
135 K.To deduce the local spin susceptibility χ i from K ave i ( K amp i ), it is necessary to evaluate the hyperfine-couplingconstant a ave ( a amp ) defined above. This is achievedby comparing K ave i ( K amp i ) with bulk spin susceptibility χ spin at room temperature where the site dependence of K ave i ( K amp i ) becomes negligible; that is, K ave i ∝ χ spin ( K amp i ∝ χ spin ). At room temperature, K ave i ≈ K amp i ≈
440 (in ppm) with little site dependences(Fig. 7) and χ spin = 6 . × − emu/mol f.u. mea-sured by Rothaemel et al . In terms of them, the hy-perfine coupling constants are evaluated as a ave ≈ . µ B and a ave ≈ . µ B . To check whetherthese coupling constants are applicable under arbitraltemperatures, we compare the temperature dependencesof P i K ave i / P i K amp i / χ spin (crosses), as depicted inFig. 8, where the summation is taken over all moleculesin the unit cell (i.e., molecules A, A, B, and C; see Fig. 1).All of these quantities are nearly scaled to each other (i.e., P i K ave i / ∝ χ spin and P i K amp i / ∝ χ spin ), which as-sure our assumptions that a ave and a amp are site- andtemperature-insensitive. Thus, it is allowed to determinethe local susceptibility χ i with these coupling constantsthrough K ave i = a ave χ i and K amp i = a amp χ i at all tem-peratures measured.In Fig. 9(a), we show the temperature dependencesof the hereby deduced local spin susceptibility χ i fromFigs. 7(a) and 7(b). [Note that χ i in Fig. 9(a) is anaverage among K ave i /a ave and K amp i /a amp because thesevalues are in good agreements with each other over theentire temperatures measured.] Susceptibility shows lit-tle site dependence around room temperature. With de-creasing temperature, however, large imbalance developsin χ i among A, B, and C nonequivalent molecules below ∼
270 K, which increases down to T CO . Below ∼ χ i ’s decrease monotonically with decreasing tem-perature. The tendency, χ C > χ A > χ B , is consistentwith the preceding works. Just above T CO , its ra-tio reaches 5:4:6 for A:B:C molecules. This large imbal-ance cannot be expected by a simple single-band picture.Notice that the temperature dependence of χ A has thesame profile as that of χ spin ; namely, χ A ∝ χ spin , as seenin Fig. 8 (circles).To highlight the temperature dependence of the spin-density disproportionation prominently, we depict therelative local spin density in Fig. 9(b), defined as h s i i = χ i / [(2 χ A + χ B + χ C ) / i -site HOMO to the conduction bandaround ε F . Imbalance develops in the relative local spindensities at the B and C sites with decreasing tempera-ture. On the other hand, the A site shows little temper-ature dependence ( h s A i ≈ χ spin ∝ χ A . At first glance, it seems to be plau-sible that the observed spin-density disproportionationcan be qualitatively understood with the semi-metallicband picture as a redistribution process of B- and C-siteHOMO contributions in the two bands. Indeed, band-structure calculations predict two bands locating in thevicinity of the Fermi level ε F . However, it is difficult toexplain the strong decrease in χ spin (Fig. 8) by the sim-ple semi-metallic framework; e.g., in the case of 2D Fermisurfaces, χ spin [ ∝ D ( ε F )] is expected to show little tem-perature dependence, since the density of states D ( ε F ) isnot sensitive to the value of ε F . Moreover, the trend ofthe spin-density disproportionation is opposite from whatis expected from the x-ray diffraction measurement [Bsite: (hole) rich, and C site: (hole) poor], which is alsoanomalous as a conventional semi-metal. As we shall seein Sec. IV A, these features should be rather regardedas consequences of the anisotropic Dirac cone realizedaround ε F , which was originally predicted under hydro-static pressures in α -I . D. Temperature dependence of the C nuclearspin-lattice relaxation rate and the electroncorrelation effects
So far, the local electronic states in α -I have beenrevealed in terms of the static susceptibilities. The nu-clear spin-lattice relaxation rate, 1 /T , probes the fluctu-ations of electron spins. In this section, we show the site-dependent spin dynamics uncovered by the site-selectivemeasurements of 1 /T T , which allows one to see the cor-relation effects in site-specific or band-specific manners,as discussed in Sec. IV B.In the B and C molecules with inversion center, the twoneighboring carbons are equivalent and give the identicalrelaxation rate. However, the two nonequivalent carbonsin the molecule A without inversion center should exhibitdifferent relaxation rates due to different hyperfine cou-pling constants as reported earlier. Nevertheless, theso-called T process, which works to average the site-specific relaxation rates among these two carbons, tendsto alter the two values toward some intermediate val-ues in-between them. Then, the distinction of the twoobserved rates is not so informative, but their average is a meaningful value specific to the A molecule. Thus,we determined the relaxation rate representative of themolecule A from the relaxation curves of the whole spec-tra (i.e., the quartet), which were nearly single exponen-tial.In Fig. 10(a), we present the temperature dependenceof the C nuclear spin-lattice relaxation rate, 1 /T , di-vided by temperature T at ψ = 110 ◦ under H//ab . Withdecreasing temperature, 1 /T T decreases monotonicallyat all sites from room temperature down to T CO . Thetemperature dependence is, however, different from siteto site. The decreasing rates are large at A and B sites,while C site exhibits only moderate temperature depen-dence. In a conventional single-band metal, 1 /T T is notexpected to show site-specific temperature dependencesbecause all sites probe the same electronic properties inthe conduction band. The distinct behaviors at A, B,and C sites indicate that the electronic structure in thissystem cannot be interpreted within a simple single-bandmodel.Since the values of 1 /T T K measures the degree ofelectron correlations in the conducting state, 1 /T T K was evaluated at each site. However, the Knight shiftat A site is too small ( K A = − ∼ −
70 ppm) at thisfiled geometry in Fig. 5(c) ( ψ = 110 ◦ ) to obtain reli-able values of 1 /T T K , compared to B and C sites. Wethus measured the spin-lattice relaxation rate 1 /T onA site at ψ = 50 ◦ , where the Knight shift at A site islarge and hence the relative error in 1 /T T K becomessmall. The results are shown in Fig. 10(b). Then, weevaluated the values of 1 /T T K for the data series at ψ = 50 ◦ for A site, and at ψ = 110 ◦ for B and C sites,respectively. In the present case of anisotropic hyper-fine couplings, and in the presence of electron correla-tions, 1 /T T K is expressed as the modified Korringarelation ; (1 /T T ) i K − i = (4 πk B / ~ )( γ n /γ e ) β i ( ζ i , η i ) K i ( i = A, B, and C). Here, γ e is the gyromagnetic ratio ofan electron, k B is the Boltzmann constant [which yield(4 πk B / ~ )( γ n /γ e ) = 2 . × sec − K − ], β i ( ζ i , η i )is the i -site correction factor for the anisotropy of thehyperfine-coupling tensor as defined in the following ( i =A, B, and C) β i ( ζ i , η i ) = ( a xxi /a zzi ) (sin η i + cos ζ i cos η i ) + ( a yyi /a zzi ) (sin η i + cos ζ i cos η i ) + sin ζ i a xxi /a zzi ) sin ζ i cos η i + ( a yyi /a zzi ) sin ζ i sin η i + cos ζ i ] , (2)and K i is the i -site Korringa ratio , which reflects the typeand strength of the electron correlations. The directionof H , ( ζ i , η i ) [see Fig. 3(b)], is determined at i = A,B, and C molecules for the present field geometries, asshown in Table II, based on the molecular orientations in Ref. 34. The principal components of the C hyperfinecoupling tensor, a xxi , a yyi , and a zzi , are determined at eachmolecule i (listed in Table II) from the total shift tensors,given in Table I, and the estimated chemical-shift tensorsat 260 K (see Sec. III C). The values of β i ( ζ i , η i ) are alsopresented in Table II.Figure 11 represents the temperature dependence ofthe Korringa ratio K i [ ∝ (1 /T T ) i K − i ]. K i = 1 meansthat there is no electron correlations. Figure 11 showsrelatively large K i for all sites ( K i ∼ − K i shows little site dependence. Uponcooling, however, it begins to exhibit distinct behaviorsfor A, B, and C molecules below ∼
270 K. The Kor-ringa ratios exhibit only small temperature dependencesat A and C sites ( K A ≈ K C ≈ − K B increases below 200 K and reaches a value of 11at 140 K. In the lowest temperature region, the mag-nitude relation of K i is given by K B > K A > K C .We note that the well-studied dimer-type organic metal κ -(BEDT-TTF) Cu[N(CN) ]Br, which resides onthe verge of the Mott transition, shows K ∼ −
10. Onthe other hand, in θ -(BEDT-TTF) I , known as a good2D metal with a 1/4-filled band, exhibits K ≈ −
3, point-ing to weak correlations. In our case, K i is intermediateor as large as in κ -type salt. This indicates the impor-tance of electron correlations in α -I , which is discussedin more details in Section IV B. IV. DISCUSSION
As we mentioned in the preceding sections (Sec-tions III C and III D), it is difficult to explain the ob-served anomalous decreases in spin susceptibility χ spin (Fig. 7), temperature-dependent spin density dispropor-tionations (Fig. 9), and the large temperature and sitedependences of spin fluctuations K i [ ∝ (1 /T T ) i K − i ](Fig. 11) in terms of simple semi-metallic or single bandpictures. Then, how can we understand our NMR re-sults comprehensively? Intriguingly, there are two hintsin the previous works. First, as we noted in Sec. I, re-sistivity shows weak temperature dependence in this sys-tem both at ambient and high pressures. Secondly,the NMR local spin susceptibilities at intermediate andhigh pressures exhibit very close behaviors to our re-sults at ambient pressure. These results suggest thatthe ambient- and high-pressure conducting states havequalitatively similar features. Furthermore, the observedspin-density disproportionation (Fig. 9) reflects the pres-ence of site-dependent potentials. The trend of the dis-proportionation is the same range as that expected inthe local site-potentials in Ref. 18, which is predictedto stabilize the Dirac cone. All of these consid-erations thus imply that the Dirac cone dominates theelectronic properties even at ambient pressure. In fact,as shall be discussed below, the observed features areproperly captured by the theoretical consequences of theanisotropic conical dispersion realized around ε F . Wewill focus on the local spin density disproportionations inSec. IV A, and the site-dependent spin fluctuation effectsin Sec. IV B.
A. Disproportionation of the local spin densitiesand the site-specific k -space anisotropy in theconduction band First, we consider the temperature and site depen-dences of the local spin densities. Let’s assume that thewave function in the conduction band is given by theBloch sum of the HOMO (highest occupied molecularorbital) at the i -site ET molecule ( i = A, B, and C), φ i [= φ ( r − R l − δ i )], as follows:Ψ k ( r ) = X R l X i =A , B , C e i k · R l C i, k φ ( r − R l − δ i ) , (3)where R l stands for the position of one α -I unit cell, and δ i is the vector connecting the i -site to the A site in theunit cell ( δ A = 0). The experimentally obtained i -sitelocal spin susceptibility χ i is proportional to the thermalaverage of φ i contributing to the conduction band around ε F , (cid:10)P k | C i, k | (cid:11) ε = ε F ± k B T , which is explicitly given as χ i = − R ∞−∞ dεD i ( ε ) f ′ ( ε ) = − R ∞−∞ dε P k | C i, k | δ ( ε − ξ k ) f ′ ( ε ), with f ( ε ) the Fermi-Dirac distribution function, ξ k the energy-momentum dispersion of the conductionband, and D i ( ε ) the local electronic density of states atmolecule i . Prime stands for the derivative with respectto ε .There are two significant consequences of the band-structure calculations relevant to the present results; one is the large tilting effect of the Dirac cone, andthe other is the resulting strong angular dependences of | C i, k | ’s about the cone. The ratio of the steepest andgentlest slants (i.e., anisotropy of the Fermi velocities) isestimated at about 10, and the latter leads to a flatband dispersion giving the van Hove singularity around10 meV above the Dirac points, as schematically de-picted in Fig. 12. Noticeably, | C B , k | is largest aroundthe steepest dispersion, denoted as S in Fig. 11, andshows a node around the gentle dispersion, denoted asG, whereas | C C , k | has opposite characteristics. | C A , k | is predicted to show no remarkable angular dependence.This means that the local spin susceptibilities at B and Csites preferentially probe the thermal excitations aroundthe S and G portions of the cone, respectively, while thesusceptibility at the A site sees the average over the cone.The overall temperature dependence of the spin suscep-tibility, shown in Fig. 8, is basically explained in thiscontext as follows: the conical dispersion has an energy-linear density of states, which gives rise to a linearlytemperature-dependent spin susceptibility. The de-creasing total susceptibility with lowering temperature,observed below 200 K in Fig. 8, can be thought of as signi-fying this. The leveling-off of the susceptibility at highertemperatures, on the other hand, implies the breakdownof the cone picture at high energies. Actually, the calcu-lated total spin susceptibility based on the band structuretends to saturate at high temperatures. The spin density disproportionation of χ B < χ C , en-hanced at low temperatures [see Figs. 9(a) and 9(b)],reasonably corresponds to the small and large local den-sity of states at the S and G portions on the cone. Thepeak formation of χ C around 190 K is attributable tothe van Hove singularity located around G, where | C C , k | shows the maximum. On the contrary, | C B , k | is van-ishingly small at G, as mentioned above, which explainswhy χ B continues to increase monotonously up to roomtemperature. Meanwhile A site shows temperature de-pendences in between B’s and C’s both in the local spinsusceptibility χ A and the relative local spin density h s A i [see Figs. 9(a) and 9(b)]. χ A scales to the bulk spinsusceptibility, χ spin , over the whole temperature range( χ A ∝ χ spin ; see Fig. 7), which is reflected in the tem-perature independent profile of h s A i shown in Fig. 9(b).These results indicate that A site probes the whole con-ical dispersion in an averaged manner, and supports thetheoretical prediction that | C A , k | shows no remarkableangular dependence on the cone. At high temperatures of hundreds Kelvin, the elec-tronic mean free path in molecular conductors can reachthe order of the unit cell size due to the strong electron-phonon scatterings. The high-temperature equalizationof the local spin susceptibilities [see Fig. 9(a)] might bepartially due to this scattering effect, which will ther-mally average the anisotropy of the wave functions inthe k -space. Several organic conductors exhibit badmetal natures such as a loss in Drude weight in opticalconductivity, or broadening in ARPES spectra at hightemperatures. Moreover, the correlation-induced ef-fect such as charge ordering is disturbed at high tem-peratures in general. In the present system, the chargedisproportionation, if it is enhanced by electron correla-tions, is expected to be depressed at high temperatures.This effect tends to make the Dirac point pushed downbelow ε F , and may lead to a crossover from the Diracsystem to a semi-metal. Actually, the spin susceptibil-ity and Korringa ratio are both temperature- and site-independent around room temperature. It is probablethat the picture of the massless Dirac electrons is brokenby these thermal effects.To illustrate how the hole density at the molecule i , ρ i ,relates to the above discussed i -site spin susceptibility χ i ,we made a comparison between ρ i and χ i . Here, the valueof ρ i is estimated from the intra-molecular bond lengthsdetermined by the X-ray diffraction study by Kakiuchi et al . and the charge-sensitive modes in the vibrationalspectroscopy by Wojciechowski et al . They found that ρ B > ρ C , which is supported by Hartree-Fock calcula-tions based on the extended Hubbard model. This re-lation is, however, opposite to the spin density profile, χ B < χ C . Because our material is a 3/4-filled electron-band (or a 1/4-filled hole-band) system (see Sec. I), thecharge (hole) density might correlate to the spin den-sity, which seems irreconcilable with the present results atfirst glance. However, the charge-spin correlation holdswhen both charges and spins are spatially well local-ized. For itinerant electron systems, on the other hand,they should not necessarily match since the amount of the hole is given by ρ i ∝ P k | C i, k | [1 − f ( ξ k )] ≡ (cid:10)P k | C i, k | (cid:11) ε ≥ ε F [with P i ρ i = 2 ( i : all moleculesin unit cell)], while the spin density is expressed as χ i ∝ (cid:10)P k | C i, k | (cid:11) ε = ε F ± k B T as mentioned above. Ob-viously, χ i is determined by | C i, k | only in the vicinityof ε F , while ρ i reflects the whole summation of | C i, k | over the conduction band above ε F . Hence, we acquirethe following relations from the experimental results: (cid:10)P k | C B , k | (cid:11) ε = ε F ± k B T < (cid:10)P k | C C , k | (cid:11) ε = ε F ± k B T and (cid:10)P k | C B , k | (cid:11) ε ≥ ε F > (cid:10)P k | C C , k | (cid:11) ε ≥ ε F , which suggestthat P k | C B , k | < P k | C C , k | holds near ε F whereas P k | C B , k | > P k | C C , k | is valid well above ε F . Thesecontrasting relations at low and high energies are consis-tent with the picture of the conical band dispersion whichis characterized by the distinct anisotropies in | C B , k | and | C C , k | ; the portion G (in Fig. 12), which is con-tributed largely from the site C, is cut off by the Brillouinzone boundary at low energies, while the portion S rele-vant to the site B extends to higher energies and over awide range of the k -space covering the zone center (seeFig. 12). B. Korringa ratios and spin fluctuation effects
Next, we turn our attentions to the site-specific spinfluctuation effects. As noted in Section III D, the Ko-rringa ratio, K i ∝ (1 /T T ) i K − i , is a measure of thestrength and type of spin fluctuations in the conductionband. As represented in Fig. 11, K i is strongly siteand temperature dependent in our system, in contrastto a simple metal where K i should be independent onsites and temperatures. This anomalous behavior is un-derstood based on the theoretical consequence that eachmolecular orbital φ i at the site i contributes to the coni-cal dispersion with a differing angular dependence in the k -space, which is consistent wi th the discussion in thepreceding section.First, it should be reminded that | C A , k | has a littleangular dependence in the reciprocal space. Hence, K A is expected to probe the fluctuations over the entire Diraccone on average and to serve as a benchmark for thestrength of correlations in the system. As we mentionedabove, K A exhibits a relatively large value ( K A ≈ /T ) A ,is given by (1 /T ) A = 2 (cid:16) γ n γ e (cid:17) k B T ~ (cid:0) a ⊥ A (cid:1) X Q χ ′′ ( Q , ω ) ω , (4)where a ⊥ A is the transverse component of the hyperfinecoupling tensor at the site A, ω is the NMR resonancefrequency, and χ ′′ ( Q , ω ) is the imaginary part of the dy-namical spin susceptibility. The wave vector, Q , andthe frequency ω are characteristic for the spin fluctua-tions associated with the electron-hole pair excitationsat ε F . For the free electrons, χ ′′ ( Q , ω ) is expressed as χ ′′ ( Q , ω ) = πγ ~ ω [ D tot ( ε F )] / with D tot ( ε F ) the to-tal density of states at ε F . In the presence of spin fluctua-tions, on the other hand, χ ′′ ( Q , ω ) are enhanced from theconstant value χ ′′ ( Q , ω ) at the wave vector Q connect-ing the degenerate states around ε F . The present systemhas two Dirac cones (at valley k D and − k D ) in the firstBrillouin zone, which are mutually connected with time-reversal symmetry. ( k D stands for the position of theDirac point inside the first Brillouin zone.) In this situ-ation, two kinds of interactions are conceivable: (i) theintra-valley scattering with Q ∼
0, and (ii) the inter-valley scattering with Q ∼ k D . The former scatteringgives ferromagnetic fluctuations while the latter leads toantiferromagnetic ones. The result of K A ≈ In contrast to | C A , k | , | C B , k | and | C C , k | have so dis-tinct angular and energy dependences on the cone that K B and K C are expected to measure the fluctuation ef-fects in preferential areas of the k -space; namely, the Sand G portions in Fig. 12, correspondingly. Note that,roughly speaking, K i probes the intensity of spin fluctua-tions per thermally excited quasi-particles instead of theintensity itself because (1 /T T ) i is divided by K i . Atlow temperatures, K B is much enhanced compared with K C ; at 140 K, the value of K B reaches 11, which is twiceof that of K C . The thermally excited states probed atB site are well located around the Dirac point in the 2D k -space because the dispersion of the S portion is steep(see Fig. 12). In the G portion, on the other hand, thethermal excitation should be extended in a wide range of k -states because of the flat dispersion with the van Hovesingularity located around 150 K above ε F . The different k -space profile of thermal excitations probed at B and Csites should give a corresponding difference in the inter-valley scattering seen at B and C sites. The inter-valleyscattering probed at B site are well restricted to the vicin-ity of Q ∼ k D , whereas that probed at C site should bespread around Q ∼ k D . The present results imply thatthe antiferromagnetic fluctuations are sharply enhancedat a wave vector of Q ∼ k D , and that the inter-valleyscattering is responsible for the electron correlations inthis system.At elevated temperatures, K B and K C gradually ap-proach each other, and all K i ’s become nearly the samearound room temperature ( K i ≈ k -space av-eraging due to the intense phonon excitations makes the difference between the molecules A, B, and C indistin-guishable, and/or the possible depression of charge dis-proportionation pushes down the Dirac point below ε F ,causing a semi-metallic state.Theoretically, electron correlations stabilize the Diraccone dispersion in this system as we mentioned inSec. I. Furthermore, the presence of the tiltedDirac cone can largely enhance the anisotropy of | C B , k | and | C C , k | in the k -space. The present site-dependentNMR characteristics well below room temperature areconsistent with these predictions and give an opportu-nity to look into the magnetic properties of the tiltedDirac cone in a k -dependent manner. V. CONCLUSION
We performed C NMR measurements on the charge-disproportionated conducting state in the layered organicconductor α -(BEDT-TTF) I at ambient pressure. Re-flecting the low crystal symmetry, we obtained separateNMR lines for crystallographically nonequivalent threemolecules in the unit cell ( i = A, B, and C). The tem-perature dependences of the resultant site-specific Knightshift K i are properly captured by the conical disper-sion with the Dirac points close to ε F . The analysesof the site-selected nuclear relaxation rate (1 /T T ) i and K i point to the presence of strong or intermediate anti-ferromagnetic spin correlations. Exploiting the theoreti-cal prediction of the peculiar site-specific reciprocal-spaceanisotropy in the Dirac cone, the present results turn outto show that the local spin susceptibility and electroncorrelations are both strongly angular dependent on thecone. This outcome is regarded as one of the outstandingaspects inherited from the tilted Dirac cone in α -(BEDT-TTF) I . VI. ACKNOWLEDGEMENTS
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Rectangular representsnonequivalent ET molecules (= A, B, and C) in the unit cell. i ( δ i ) xx (ppm) ( δ i ) yy (ppm) ( δ i ) zz (ppm)A 48 77 869B 51 47 780C 59 130 929TABLE I: The µ -axis component of the C-hyperfine-shifttensor δ µµi ( µ = x , y , z ; see Fig. 3) at molecule i ( = A, B,and C) deduced from the angular dependences of the NMRshift δ i ’s at 260 K shown in Figs. 4(a) and 4(b). (2009). K. Takenaka, M. Tamura, N. Tajima, H. Takagi, J. Nohara,and S. Sugai, Phys. Rev. Lett. , 227801 (2005). K. Koizumi, K. Ishizaka, T. Kiss, M. Okawa, R. Kato, andS. Shin, private communications. T. Moriya, J. Phys. Soc. Jpn. , 516-520 (1963); A.Narath and H. T. Weaver, Phys. Rev. , 373-382 (1968). FIG. 2: (Color online) (a), (b) Angular dependences of the C NMR line shapes under (a)
H//ab and (b)
H//bc at 260K. The definitions of angles ψ and θ are given in the insets. [Note that the origins of ψ and θ are set as H//b and
H//c ,correspondingly]. (c), (d) Nuclear dipolar splitting width d at A, B, and C nonequivalent molecules extracted from Figs. 2(a)and 2(b), respectively. Calculated angular dependences are shown together based on Ref. 34 (curves). In all figures, symbolsrepresents the data relevant to molecules A (open circles), B (triangles) and C (crosses), respectively. i a xxi (kOe/ µ B ) a yyi (kOe/ µ B ) a zzi (kOe/ µ B ) ζ i (degree) η i (degree) β i ( ζ i , η i )A -1.11 -1.61 13.35 21.4 57.2 0.106B -1.26 -2.25 11.91 42.6 67.7 1.139C -0.74 -0.70 14.32 31.8 58.8 0.281TABLE II: Principal components of the hyperfine-coupling tensor a µµi ( µ = x , y , z ) at molecule i , spherical polar angles ( ζ i , η i ) [defined in Fig. 3(b)], and the correction factors for the anisotropy of the hyperfine-coupling tensor β i ( ζ i , η i ) [see Eq. (2) inSec. III D] at i = A ( ψ = 50 ◦ ), B ( ψ = 110 ◦ ) and C ( ψ = 110 ◦ ) molecular sites. FIG. 3: (Color online) (a) Molecular principal axes ( x , y , z )of ET molecule. (b) Spherical polar coordinate ( ζ i , η i ) of theexternal field vector H at i -site ET molecule ( i = A, B, andC) in the principal axes ( x , y , z ). FIG. 4: (Color online) Angular dependence of the C NMRcentral shift at molecule i (= A, B, and C), δ i , under (a) H//ab and (b)