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[Yugo Oshima](https://orcid.org/0000-0001-9822-8262), [Yasuyuki Ishii](https://orcid.org/0000-0001-6520-3894), [Francis L. Pratt](https://orcid.org/0000-0002-5919-3885), [Isao Watanabe](https://orcid.org/0000-0003-4722-8654), [Hitoshi Seo](https://orcid.org/0000-0003-4907-7255), [Takao Tsumuraya](https://orcid.org/0000-0001-9063-9278), [Tsuyoshi Miyazaki](https://orcid.org/0000-0003-3534-4404), [Reizo Kato](https://orcid.org/0000-0002-2606-4657)

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[Quasi-One-Dimensional Spin Dynamics in a Molecular Spin Liquid System](https://mdr.nims.go.jp/datasets/457b26f1-3a4d-49bb-ae38-94948d48a463)

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arXiv:2410.20076v1  [cond-mat.str-el]  26 Oct 2024Quasi-one-dimensional Spin Dynamics in a Molecular Spin Liquid SystemYugo Oshima,1, ∗ Yasuyuki Ishii,2 Francis L. Pratt,3 Isao Watanabe,4Hitoshi Seo,1, 5 Takao Tsumuraya,6 Tsuyoshi Miyazaki,7 and Reizo Kato11RIKEN Cluster for Pioneering Research, Wako, Saitama 351-0198, Japan.2College of Engineering, Shibaura Institute of Technology, Minuma-ku, Saitama 337-8570, Japan.3ISIS Neutron and Muon Source, STFC Rutherford Appleton Laboratory, Chilton, Oxfordshire OX11 0QX, UK4RIKEN Nishina Center, Wako, Saitama 351-0198, Japan.5RIKEN Center for Emergence Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan.6Magnesium Research Center, Kumamoto University, Kumamoto 860-8555, Japan7Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan(Dated: October 29, 2024)The molecular triangular lattice system, β’-EtMe3Sb[Pd(dmit)2]2, is considered as a candidatematerial for the quantum spin liquid (QSL) state, although ongoing debates arise from recent con-troversial results. Here, the results of electron spin resonance (ESR) and muon spin relaxation (µSR)measurements on β’-EtMe3Sb[Pd(dmit)2]2 are presented. Both results indicate characteristic behav-iors related to quasi-one-dimensional (q1D) spin dynamics, whereas the direction of anisotropy foundin ESR is in contradiction with previous theories. We succeed in interpreting the experiments bycombining density-functional theory calculations and analysis of the effective model taking into ac-count the multi-orbital nature of the system. While the QSL-like origin of β’-EtMe3Sb[Pd(dmit)2]2was initially attributed to the magnetic frustration of the triangular lattice, it appears that theprimary origin is a 1D spin liquid resulting from the dimensional reduction effect.Ground states of magnetically frustrated triangu-lar lattice systems, where antiferromagnetically coupledS=1/2 spins are positioned on each lattice site, haveposed a long-standing challenge in condensed matterphysics since Anderson proposed the resonating-valence-bond (RVB) state for the S=1/2 Heisenberg model on auniform triangular lattice [1–7]. Although confirmationof the quantum spin liquid (QSL) state has remainedelusive, several molecular materials with S=1/2 trian-gular lattices have garnered significant attention [8–15].One notable QSL candidate is the dimer-Mott insula-tor β’-EtMe3Sb[Pd(dmit)2]2, where dmit, Et and Me are1,3-dithiole-2-thione-4,5-dithiolate, ethyl and methyl, re-spectively [13–16].β’-EtMe3Sb[Pd(dmit)2]2 consists of Pd(dmit)2 anionsand a monovalent countercation EtMe3Sb+. The an-ions form strongly dimerized [Pd(dmit)2]−2 , and an elec-tron is localized on each dimer. As shown in Fig. 1,the crystal structure of β’-EtMe3Sb[Pd(dmit)2]2 has twocrystallographically equivalent Pd(dmit)2 layers with dif-ferent dimer stacking direction (layer A and B). Lay-ers A and B are connected by the glide plane symme-try. The Pd(dmit)2 dimers form a S=1/2 triangular lat-tice on each layer, where the transfer integral along thedimer’s stacking direction is denoted as tB, the side-by-side direction as tS , and the diagonal direction as tr (Fig.1). The calculated transfer integrals between the dimerssuggest a nearly isosceles or scalene triangular arrange-ment in the magnetic geometry (see Table I) [17–22]. β’-EtMe3Sb[Pd(dmit)2]2 shows no magnetic long-range or-der down to approximately 40 mK, which is several ordersof magnitude lower than its exchange interaction [23–33].Precise tuning of the transfer integrals using mixed coun-tercations further reveals the presence of a QSL ‘phase’in the system [34].However, the nature of its QSL-like ground state isso far controversial. The specific heat and the first re-port of thermal conductivity measurements both showlinear temperature dependence, although it is an insu-lator [24, 25]. Such behavior is considered to be origi-nating from the delocalized nature of the excitations andit was proposed to be owing to the existence of spinonswith a Fermi surface. However, theoretical studies for aQSL state with a spinon Fermi surface find T 2/3 scalingrather than linear scaling [35]. Furthermore, in contra-diction to the initial study of thermal conductivity, re-cent studies performed by three different groups reportthe absence of any residual linear term κ0/T [29, 30, 33],while Yamashita et al. have reported that the linearterm depends on the sample and in particular the cool-ing rate [31, 32, 36]. No cooling rate dependence is ob-served in X-ray diffraction, transport and NMR, though[37]. As for alternative ground states, several compet-ing charge-order states are proposed through vibrationalspectroscopy [28]. A random-singlet state due to ran-dom intradimer charge disproportionation is proposedsince a relaxor-type ferroelectricity was observed [38–40]. Moreover, a recent ab-initio calculation proposedthat the QSL state of β’-EtMe3Sb[Pd(dmit)2]2 is essen-tially a 1D spin-liquid, although it preserves some two-dimensionality [22].In this Letter, we present a different experimen-tal approach for studying the ground state of β’-EtMe3Sb[Pd(dmit)2]2 using electron spin resonance(ESR) and muon spin rotation (µSR). Both results ex-hibit characteristic features of a quasi-one-dimensionalhttp://arxiv.org/abs/2410.20076v12tB trtSa BbLayer AθtBtStrbLayer AtSttLayer BBabθtBtStr!!acbPd(dmit)  dimer2(S=1/2)FIG. 1. (left) Crystal structure of β’-EtMe3Sb[Pd(dmit)2]2.Two crystallographically equivalent Pd(dmit)2 layers with dif-ferent dimer stacking directions exist in the unit cell (Layer Aand B), and the Pd(dmit)2 dimer with S=1/2 forms a trian-gular lattice in each layer. (right) Schematic drawings of thetriangular lattice and its transfer integrals for layers A and B.θ is the angle of the magnetic field from the a-axis used forESR.TABLE I. Interdimer transfer integrals (in meV) alongthe three directions of the triangular lattice of β’-EtMe3Sb[Pd(dmit)2]2 evaluated by different methods. Thelow-temperature crystal structure data has been used for ourcalculation. FP, TB and EHM stands for first-principles cal-culation, tight-binding and extended Hubbard models, respec-tively. The number of bands included in the analysis is indi-cated.Methods tB tS tr Ref.Extended Hückel 34 33 26 [17]FP + TB (6-band) 54 45 40 [18]FP + TB (2-band) 49 45 37 [19]FP + TB (2-band) 55 47 39 [20]FP + TB (2-band) 57 45 40 [21]FP + TB (2-band) 57 45 40 [22]FP + TB (8-band) + EHM 31 28 36 present study(q1D) spin dynamics, with the fastest propagation direc-tion for spin dynamics being along tr, which correspondsto the direction of weakest magnetic coupling, as indi-cated by previous calculations [17–22]. By extending thetheoretical analysis to include the multi-orbital nature ofthe system, we show a renewed picture of the magneticanisotropy, finding good agreement with the experimen-tal results. Our finding suggests the QSL ground state ofβ’-EtMe3Sb[Pd(dmit)2]2 is another example of ‘dimen-sional reduction’ induced by frustration and quantumfluctuations.Single crystals of β’-EtMe3Sb[Pd(dmit)2]2 were pre-pared using the method described in Ref. [30]. ESRmeasurements were carried out with a conventionalX-band ESR spectrometer (∼ 9.1 GHz), using a singlecrystal (approximately 1 × 1 × 0.05 mm3) mounted ona quartz rod to allow rotation in the ab-plane. The µSRexperiments were performed at the RIKEN-RAL muonfacility in the UK and the SµS facility in Switzerland.Randomly oriented crystals, with a total weight of 100mg, were wrapped in a packet of 12.5 µm silver foil andattached to the sample plate of a helium dilution refrig-erator. Further details of the experimental setups andtheoretical calculations can be found in the SupportingInformation (SI) [41].Two distinct ESR signals are observed when the mag-netic field is rotated within the ab-plane of the trian-gular lattice of the β’-EtMe3Sb[Pd(dmit)2]2 salt. Theobserved ESR spectra are presented in Fig. S2 [41]. Theangular dependence of the g-values, obtained from theESR signals, shows two almost identical components ofthe g-tensor with a shift between them of about 30◦ (Fig.2(a)). This g-tensor shift coincides with the difference inthe stacking direction of the Pd(dmit)2 dimers betweenadjacent layers, as shown in Fig. 1. Hence, the principalaxes of the g-tensor are related to the orientation of thePd(dmit)2 dimers (S=1/2), and the minimum and themaximum of the g-value are observed when the magneticfield is respectively applied parallel and perpendicularto the stacking direction of the Pd(dmit)2 dimers. Theperpendicular direction is close to the b-axis side-by-sidedirection of the dimer arrangement. From the compar-ison of the g-values with the crystal axes, the two ESRsignals can be assigned to layer A and layer B of thePd(dmit)2 layers, shown as open and solid red circles inFig. 2(a), respectively. These results show that the ESRorigin is purely from the spins on the triangular lattice,and extrinsic effects from impurities are ruled out.It is well known that if a finite exchange interaction ex-ists between two spins with different g-tensors, the twoindependent ESR absorption lines merge into a singleabsorption line in a process known as exchange narrow-ing [42, 43]. The exchange interaction J can be roughlyestimated from the relation 2J ∼ |∆g|µBB, where ∆gis the difference in the g-values when the amalgama-tion of ESR lines occurs. [42, 44] From Fig. 2(a),we observe that the amalgamation occurs where ∆g =0.005, leading to an estimated interlayer exchange in-teraction of approximately 0.54 mK. This small inter-layer interaction suggests that the magnetic network ofβ’-EtMe3Sb[Pd(dmit)2]2 is highly 2D, which is consis-3br At"#$"%$1.9822.022.042.062.082.1-50 0 50 100 150 200Layer ALayer Bg-ValueAngle  (deg)b-axisa-axisStacking direction (tB)Side-by-side (tS)a-axist t04812-50 0 50 100 150 200Angle  (deg)Linewidth (mT)(tB)Layer BStacking(tB)Diagonal (tr)Side-by-side (tS)04812a-axis b-axis a-axis(tr)Stacking     (tB)Layer ASide-by-side (tS)Diagonal (tr)FIG. 2. The angular dependence of (a) the g-value and (b)the ESR linewidth of β’-EtMe3Sb[Pd(dmit)2]2 at 4.7 K wherethe magnetic field is rotated within the ab-plane (a-axis is atθ=0◦). Two ESR signals from layers A and B are presented asred open and solid circles, respectively. The angular depen-dence of the linewidth is fitted with the sum (thick blue curve)of 1+cos2 (θ − θgmax) and (3 cos2 (θ − θq1D)−1)2 terms, pre-sented as orange and green solid curves, respectively. θg max is104◦ and 76◦ for layers A and B, respectively, which is in goodagreement with the angle dependence of the g-tensors. θq1Dis found to be 149◦ and 34◦ for layers A and B, respectively,which approximately corresponds to the diagonal direction trof the triangular lattice in each layer.tent with the absence of long-range magnetic order inthis system.The in-plane angular dependence of the ESR linewidthat 4.7 K obtained from ESR of layers A and B are pre-sented, respectively, as open and solid circles in Fig.2(b). The linewidth shows an unconventional angu-lar dependence, not previously seen in the case of alow-dimensional system or an inorganic triangular lat-tice [43, 45]. We find that this unusual angular de-pendence is well-fitted by a sum of (1 + cos2 θ) and(3 cos2 θ− 1)2 terms, which are presented respectively asorange and green solid curves in Fig. 2(b). The formerangular dependence is a typical one for angle-dependentESR and originates from the contributions of magneticanisotropies to the g-values and dipolar or hyperfine in-teractions. For each layer, the extrema of the (1+cos2 θ)FIG. 3. Magnetic field dependence of the relaxation rate λof β’-EtMe3Sb[Pd(dmit)2]2 at (a) 28 mK, (b) 0.85 K, and(c) 2.0 K. The red solid lines and the blue dashed lines arethe best fits from the q1D model and the 2D diffusive model,respectively. (d) Temperature dependence of intra-chain dif-fusion rate D‖.term coincide with those of the g-values (tB and ⊥ tB,respectively). The (3 cos2 θ− 1)2 angular term originatesfrom q1D spin diffusion, which is commonly observed inESR studies of low dimensional spin systems [43, 46–49]. For q1D spin diffusion, the direction showing themaximum of the linewidth corresponds to the diffusiondirection. Our results show that the diffusive direction isalong the diagonal direction of the triangular lattice (tr)for both layers A and B. Despite our system consistingof a 2D triangular magnetic network, the ESR resultsindicate a q1D spin dynamics along tr.Next, we present our µSR results for β’-EtMe3Sb[Pd(dmit)2]2. Neither a spontaneous precessionsignal nor a divergence of the muon depolarization rate λwas observed down to low temperatures under zero-fieldconditions (ZF-µSR, Fig. S3(a) in SI) [41]. These resultsshow that there is no sign of long-range order downto 28 mK, in good agreement with the specific heatand thermal conductivity measurements [24, 25, 29–32].Moreover, no excitation gap is found from the fielddependence of the transverse-field muon spin rotation(TF-µSR) measurements (Fig. S4).To gain information from the spin dynamics, the fielddependence of λ has been studied. Longitudinal-fieldswere applied along the muon-spin polarization direction(LF-µSR) and the field-dependent muon-spin depolariza-tion rate λ(B) was measured at various temperatures(Figs. 3(a)-(c)). Each one shows a B−0.5 dependence4over a wide field range of 0.1 < B < 100 mT (red linein Figs. 3(a)-(c)) at temperatures below 2 K. This char-acteristic behavior reflects a 1D spin diffusion, which isfully in accordance with the ESR results. Note that ourresults cannot be fitted with a 2D diffusive model (bluedashed lines). From these field dependences, we couldobtain diffusion rates D‖ using the expressionλ(B) =A24(2D‖γeB)−1/2 (1)where A is a scalar hyperfine coupling constant and γeis a gyromagnetic ratio of electron. Its derivation can befound in the SI [41]. Our DFT calculations for muon ad-ditions to Pd(dmit)2 using Gaussian16 [50] show the low-est energy when the muons were added to the end S site ofPd(dmit)2 molecule, with the corresponding experimen-tal value of 71 MHz for A [41]. This value is comparablewith the value of 82 MHz found for muonium additionat the end of the electron acceptor molecule TCNQ [51].Using the obtained coupling constant, the diffusion rateD‖ is estimated as being of the order of 1012 s−1. Thetemperature dependence of D|| is shown in Fig. 3(d). Wecan also evaluate the degree of one-dimensionality fromthe data, giving a lower limit estimate of the ratio ofintra-chain to inter-chain diffusion rate as D‖/D⊥ > 104[41]. This implies that the diffusion is highly 1D despitethe three transfer integrals around the triangular unithaving rather similar values, according to calculation.The µSR and ESR results reveals a gapless groundstate with a q1D spin dynamics. This might suggestthe QSL-like ground state of β’-EtMe3Sb[Pd(dmit)2]2 isessentially a 1D spin-liquid. However, the diffusive direc-tion is found to be along tr, which is the smallest transferintegral from previous theoretical estimations as shownin Table I [17–22].Following these results, we have reanalyzed its elec-tronic structure (see Ref. [41] for details). Using first-principles calculations based on the density functionaltheory (DFT), we derive the maximally-localized Wan-nier functions (MLWFs) from 8 bands near the Fermilevel. Here, following recent studies [21, 53], plane-wave DFT calculation within the generalized gradientapproximation [54] was performed with the QUANTUMESPRESSO code [55] using norm-conserving pseudopo-tentials [56, 57]. MLWFs were generated using the Wan-nier90 package [58, 59], by setting a Wannier center ateach dmit ligand. Note that these MLWFs form notonly the highest occupied molecular orbital (HOMO) butalso the lowest occupied molecular orbital (LUMO), andeight bands are formed from them since there are fourPd(dmit)2 molecules in the unit cell. In Fig. 4(a), theDFT band structure is shown together with the tight-binding (TB) bands based on the MLWFs, whose spatialforms are shown in Fig. 4(b). The MLWFs are dis-tributed on either side of the molecule, that show similar-ity to the wave functions introduced and called fragmentFIG. 4. (a) Band structure of β’-EtMe3Sb[Pd(dmit)2]2 (bluesolid curves), and the TB bands based on the MLWFs(red broken curves). (b) Two independent MLWF in thePd(dmit)2 dimer, drawn using VESTA [52]. (c) Mean-field en-ergies of different AF patterns varying the intersite Coulombenergies scaled by V0. The onsite Coulomb repulsion is setto U=1.0 eV. The AF alignments for each stable pattern areshown in the inset.molecular orbital (fMO) in Ref. [60].Based on the derived TB parameters, we investigatethe electron correlation effect by the extended Hubbardmodel and compare the mean-field energies of differentantiferromagnetic (AF) patterns. Fig. 4(c) shows the re-sult for onsite Coulomb repulsion U = 1.0 eV, and vary-ing the intersite Coulomb energies scaled by V0. One cansee that AF1 and AF3 states are competing in energy, inwhich both patterns are AF along the tr direction. Thissuggests that this direction indeed shows the strongestmagnetic interaction.We can evaluate the effective interdimer transfer inte-grals based on these calculations, following Ref. [60].The wave function where S = 1/2 is localized is de-scribed by the linear combination of the fMOs, and theirweights can be adopted from the mean-field solutionwhich is about 1:3 within the molecule, consistent withthe NMR measurement [61]. Typical parameters resultin the values listed in Table I: Intriguingly, tr becomesthe largest. This difference from previous studies comes5from the multi-orbital nature; most studies have focusedon the half-filled valence bands based on HOMO. How-ever, the HOMO-LUMO levels in the isolated Pd(dmit)2molecule are close in energy with a separation of about0.5 eV, while the Coulomb energy is of the same order[17, 20, 22, 62]. It is then natural to consider the multipleorbitals, and we have shown its significance in this study.The competition and the fluctuation of the AF1 andAF3 states might suggest that the dimensional reduc-tion effect takes place in the frustrated triangular lat-tice. In a similar manner to the triangular lattice sys-tem Cs2CuCl4, the magnetic frustration significantly re-duces the interchain correlations in the ground state, and1D physics similar to the spin-chain system can appear[3, 63–65]. Therefore, we conclude that the ground stateof β’-EtMe3Sb[Pd(dmit)2]2 might be a 1D spin liquidrather than a 2D QSL state of the triangular lattice.Cs2CuCl4 exhibits dimensional reduction within anisosceles triangular magnetic lattice where J’/J ∼ 0.4,and a small interlayer coupling stabilizes the 3D mag-netic order below TN=0.6 K [3, 63–65]. In contrast, β’-EtMe3Sb[Pd(dmit)2]2 shows dimensional reduction evenwith the larger ratio J ′/J ∼ 0.7 (from the ratio of t2values), without any sign of long-range order. Such sig-nificant dimensional reduction and the lack of long-rangeorder cannot be explained by a simple S=1/2 Heisen-berg model, and other factors, such as charge and orbitaldegrees of freedom and the infinitesimal interlayer inter-action, which causes AF instability, should be taken intoaccount. Let us note that the range of J ′/J where di-mensional reduction and gapless excitation are theoreti-cally observed remains controversial. For example, RVBtype theories suggest 0 ≤ J ′/J ≤ 0.25 in Ref. [66] and0 ≤ J ′/J ≤ 0.65 in Ref. [67]; the latter may give theupper bound. Further development of such theoreticalapproach might be able to account for the dimensionalreduction we observe at the larger value of J ′/J ∼ 0.7. Itis possible that the cation’s orientational disorder in β’-EtMe3Sb[Pd(dmit)2]2 also contributes to the AF insta-bility [16]. However, the cation disorder does not seem toaffect the magnetic network, as the diffusion anisotropyratio remains highly 1D (D‖/D⊥ > 104). Experimen-tally, the absence of long-range magnetic order in thissystem appears to be primarily determined by the in-finitesimal interlayer interactions.Muon site calculations were carried out using theSTFC SCARF compute cluster. 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