Charge and spin interplay in a new spin liquid candidate BEDT-TTF-based organic Mott insulator
Natalia Drichko, Shiori Sugiura, Minoru Yamashita, Akira Ueda, Shinya Uji, Nora Hassan, Yoshiya Sunairi, Hatsumi Mori, Elena I. Zhilyaeva, Svetlana Torunova, Rimma N. Lyubovskaya
CCharge and spin interplay in a new spin liquid candidate BEDT-TTF-based organicMott insulator
Natalia Drichko,
1, 2, ∗ Shiori Sugiura,
3, 4
Minoru Yamashita, Akira Ueda,
2, 5
Shinya Uji,
3, 4
Nora Hassan, Yoshiya Sunairi, Hatsumi Mori, Elena I. Zhilyaeva, Svetlana Torunova, and Rimma N. Lyubovskaya Institute for Quantum Matter and Department of Physics and Astronomy,Johns Hopkins University, Baltimore, MD 21218, USA The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan National Institute for Materials Science, Tsukuba, Ibaraki 305-0003, Japan Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan Department of Chemistry, Faculty of Advanced Science and Technology,Kumamoto University, Kumamoto 860-8555, Japan Institute of Problems of Chemical Physics RAS, Chernogolovka, Moscow region, 142432 Russia (Dated: February 24, 2021)Triangular lattice quasi-two-dimensional Mott insulators based on BEDT-TTF molecule and itsanalogies present the largest group of spin liquid candidates on triangular lattice. It was showntheoretically that spin liquid state in these materials can arize from a coupling to the fluctuatingcharge degree of freedom. In this work we discuss magnetic properties of one of such materials, κ -(BEDT-TTF) Hg(SCN) Cl, which is known to be at the border of the phase transition from Mottinsulator into a charge ordered state, and demonstrates charge order properties in the temperaturerange from 30 to 15 K. Our magnetic susceptibility and cantilever magnetisation measurementsdemonstrate an absence of spin order in this material down to 120 mK. We present argumentsdemonstrating that the charge order melting at low temperatures prevents ordering of spins.
INTRODUCTION
Research in frustrated magnetism is already for sometime focused on a search for a quantum spin liquid.This quantum disordered spin state is expected in sys-tems with frustrated or competing interactions, and canbe heavily influenced by disorder in the candidate sys-tems [1, 2]. Quasi-two-dimensional (2D) Mott insulatorsbased on the organic molecule BEDT-TTF [3] producedthe largest number of organic spin liquid candidates [4–9] presumably due to a presence of a ring exchange forS=1/2 system on anisotropic triangular lattice [10, 11].Another path to a spin liquid state was suggested fororganic Mott insulators, where the driving force are on-site fluctuating electrical dipole moments coupled to on-site S=1/2 degree of freedom [12–14]. These fluctuat-ing dipole moments, and associated with them quantumdipole liquid state, were detected experimentally [15, 16].Studies of magnetic effects that can be brought about bya fluctuating charge degree of freedom coupled to spinsin a Mott insulator are of high interest.In this work we probe experimentally magnetic prop-erties of κ -(BEDT-TTF) Hg(SCN) Cl ( κ -HgCl), whichat low temperatures presents is a good model of a Mottinsulator possessing an electrical dipole degree of free-dom. In κ -HgCl structure, (BEDT-TTF) dimers forma triangular lattice in the BEDT-TTF-based layer, withan average charge of one hole per (BEDT-TTF) andS=1/2 achieved by a charge transfer between anion andcation layers (Fig. 1, 3a). A metal at high temperatures, κ -HgCl undergoes a charge order metal-insulator transi-tion at 30 K, and then a continuous melting of this charge order below 15 K. This behavior allows us to study cou-pling between charge (electric dipole) and spin degrees offreedom.Melting of the charge order can lead to inhomo-geneities, appearance of domain walls, and charge dis-order. An interplay of disorder and low-dimensionality isan important direction of research for spin liquid mate-rials. One of the recent examples on a triangular latticeis YbMgGaO , where structural disorder controls mag-netic state [17]. Many materials, of which YbMgGaO is an example, posses intrinsic structural disorder. Inorder to study disorder effects in some situation disor-der is introduced by X-ray irradiation [18]. κ -HgCl pro-vides a unique situation where charge inhomogeneitiesare present only in a temperature range below 15 K, andtheir scale is temperature dependent. It can give an ad-ditional insight of how inhomogeneities control magneticstate of the system.The behavior of the charge degree of freedom in κ -HgCl is well studied by now. On the charge order tran-sition, a small charge difference of ∆ n =0.2 e betweencharge-poor and charge rich molecules of a dimer leadsto a dipole solid state [19, 20]. This charge distribu-tion in the insulating state is schematically indicatedby red and blue color for the (BEDT-TTF) layer inFig. 2. The charge order transition drives a large in-crease of d.c. resistivity [19, 21] and a notable fea-ture in heat capacity (Fig. 2) [16, 20]. While the lat-tice response is detected on this first order phase tran-sition [20], and some lattice phonons change [15], thelattice change has not been detected by the XRD studieswhich were performed so far [19]. Optical, Raman, and a r X i v : . [ c ond - m a t . s t r- e l ] F e b (a) (b) FIG. 1. (a) A projection of the structure of (BEDT-TTF) layer of κ -(BEDT-TTF) Hg(SCN) Cl on ( b, c ) crys-tallographic plane. Dashed line circles indicate dimers of(BEDT-TTF) . (b) A schematic representation of the 2Dlayer of (BEDT-TTF) dimers. Thick black lines representBEDT-TTF molecules bound in a dimer by a shared orbital(grey oval). In both (a) and (b) the red triangle show a unitof triangular lattice. length change measurements suggest, that on the order-ing transition the electronic system changes dimensional-ity from 2D to 1D due to the formation of charge stripesalong the c axis [16, 19, 20]. In accord with the experi-ments, calculations by Ref. [22] suggest that the chargeorder in κ -HgCl results in a high in-plane anisotropy ofmagnetic exchange interactions J , leading to an effec-tively 1D magnetism for κ -HgCl in contrast to κ -(BEDT-TTF) Cu (CN) . 1D antiferromagnetic (AF) stripes canresult in a 1D spin liquid, presenting another possibleway to reach a spin liquid state for κ -HgCl [23, 24]. Al-ternatively, a strong coupling of spin degrees of freedomto the lattice can result in a formation of a spin sin-glet state [25], as observed, for example, in θ -(BEDT-TTF) RbZn(SCN) [26].Here we present our results on heat capacity, SQUIDmagnetic susceptibility, and cantilever torque magne-tometery measurements of κ -HgCl. As expected for acompound with a complicated phase diagram related tothe charge degree of freedom, magnetic properties of κ -HgCl also show complex temperature dependence. Themain result of our work is that no magnetic order is de-tected in this system down to at least 120 mK, while theexchange interactions are on the order of at least 100 K.In contrast to the majority of organic triangular latticespin liquid candidates, γ linear term is heat capacity isnegligibly small in this system. This points on a newpath to a spin liquid state in κ -HgCl.On metal-insulator transition in κ -HgCl at T=30 K,heat capacity shows a distinct peak (see Fig. 2 (a)). Noother phase transition is detected down to 100 mK, theextrapolation down to 0 K suggests a negligible linearcomponent of γ term in heat capacity C p = βT + γT .Magnetic susceptibility χ M ( T ) of a polycrystal sam-ple of 2.451 mg measured at 1 T (Fig. 2 (b)) revealsa complex temperature dependence. Above the metal-insulator transition, κ -HgCl shows Pauli susceptibilityof about 4 · − emu/mol, which is a value close to theother BEDT-TTF-based organic conductors [27]. On the experimental data Bonner-Fisher c M ( e m u / m o l ) * - Temperature (K)Temperature (K) C p ( m J / m o l e K ) * CO insulator metalCO melting (a) (b)
FIG. 2. (a)Temperature dependence of specific heat of κ -(BEDT-TTF) Hg(SCN) Cl at 0 T. (b) Temperature depen-dence of χ M ( T ) (points), green line is magnetic susceptibilityof a Bonner-Fisher model with J =200 K and 1 e per dimer.Three regimes related to the charge degree of freedom [16] areindicated by color and the schemes of charge distribution onthe orbital of (BEDT-TTF) dimer, where grey indicates ho-mogeneous charge, red is charge-rich, and blue is charge-poor. metal-insulator transition at 30 K magnetic susceptibility χ M ( T ) does not show any change within the noise of themeasurements. Instead, magnetic susceptibility starts todecrease abruptly on cooling below about 24 K. However,instead of decreasing down to the lowest values with χ M = 0, as is expected in case of a spin gap or an anti-ferromagnetic ordering, magnetic susceptibility starts torise again on cooling the sample below 15 K. χ M ( T ) sat-urates below about 5 K, with the saturation values ofabout 3.5 · − emu/mol. No indication of magnetic or-dering is found in χ M ( T ) of κ -HgCl measured down to2 K.BEDT-TTF based crystals are typically very small,and posses very low magnetic susceptibility (Fig 2). Inorder to detect possible magnetic ordering or singlet for-mation in the charge ordered state, and to understandthe nature of the low temperature magnetic state, weperformed measurements of cantilever torque magnetiza-tion for single crystals of κ -HgCl. This method proved tobe the most sensitive to detect magnetic order, and wassuccessfully applied to organic Mott insulators [28–30].Magnetic torque signal measured for κ -HgCl is describedwell by the following equation: τ = τ + τ θ sin( θ − θ ) + τ θ sin2( θ − θ ) (1)In κ -HgCl torque response, τ sin ( θ − θ ) componentdoes not change with applied magnetic field at all mea-sured temperatures (see Supplement Information). Weconclude that it fully corresponds to the gravity force,no ferromagnetic component of torque was detected. The τ θ sin (2( θ − θ )) component in the torque response of κ -HgCl corresponds to the paramagnetic behavior.Cantilever torque magnetization measurements showthe persistence of paramagnetic response when κ -HgClis cooled through the charge order transition at 30 Kand below this temperature, but detect an abrupt in-crease of torque amplitude τ θ at T=30 K, as shown inFig. 3. Torque amplitude τ θ for the rotation in ab planeat 1 T roughly follows the temperature dependence ofmagnetic susceptibility, with a decrease at about 20 K,and an increase below 15 K. The phase of 2 θ for rotationin ac plane follows this temperature behavior, while theamplitude stays constant on cooling. These effects areweak, and are suppressed at 3 T and higher fields. -50 0 50 100 150 200 t q ( a . u . ) q (deg)
25 K 29 K 32 K 5 K 10 K 15 K 20 K
Temperature (K) q ( deg )
1T 3T 5T0 10 20 30 40-30-20-1001020 q ( deg ) Temperature (K)
1T 3T 5T ac - planeab - plane ab - planeac - plane
H-field H (a) (b) (c) (d) t s i n2 q / H ( a . u . )
1T 3T 5T 0 10 20 30 40 t s i n2 q / H ( a . u . ) Temperature (K)
1T 3T 5T
FIG. 3. (a) Scheme of cantilever torque magnetometery ex-periment with rotation direction for ab plane shown with bluearrow, and magnetic field direction shown in black. (b) Angledependence of paramagnetic component τ θ at 5 T, rotationin ab plane for temperatures related to the different chargestates of κ -HgCl. (c)-(d) Temperature dependence of an am-plitude and phase of paramagnetic component of magnetictorque τ θ , (c) shows a temperature dependence of param-agnetic torque amplitude τ θ in ac plane and ab planes. (c)shows temperature dependence of the phase θ in ac and ab planes. Magnetization was measured by following the torqueamplitude dependence on magnetic field H at an anglewhere the amplitude of torque is maximum, see Fig. 4.At temperatures T=20, 10, 5, 1.9 K for the filed up to H = 5 T torque τ shows parabolic dependence τ ∝ H ,
120 mK t m ag ( a . u . ) q (deg) steps 0.7 T t q ( f = o ) ( a . u . ) Magnetic field (T)20 K step 0.2 T 0246 1.9 K step 0.7 K 120 mK step 0.7 K t= const*H t q ( a . u . ) FIG. 4. Magnetization of κ -(BEDT-TTF) Hg(SCN) Cl mea-sured my magnetic cantilever torque magnetometery. (a)Magnetic field dependence of torque signal at 120 mK formagnetic fiel H from ) to 17.5 T. (b) Field dependence of τ θ . suggesting a paramagnetic state and absence of magneticorder. To probe the low temperature magnetic statewe performed cantilever torque magnetometery measure-ments down to 120 mK and up to 17.5 T (Fig. 4), whichconfirmed τ ∝ H behavior at this temperature up tothe highest measured field.First, we will discuss magnetic properties of κ -HgClaround the temperature of the charge order phase tran-sition. An absence of a change of magnetic susceptibilityat T=30 K is unexpected: The character of χ M ( T ) is ex-pected to change from Pauli to a behavior of an insulatorwith unpaired spins or a spin singlet state. The absenceof a change in χ M ( T ) is in contrast to the behavior ofmagnetic susceptibility observed on charge order metal-insulator transitions in other 2D BEDT-TTF-based ma-terials, which showed a formation of a spin singlet in thecharge ordered insulating state [26, 31]. The paramag-netic behavior below the transition temperatures is con-firmed by the cantilever magnetic torque measurements.The jump of the amplitude of paramagnetic torque τ θ at 30 K detects a change of the anisotropy of g-factor,which can occur due to a crystal or electronic structurechange [32]. The structural change on the charge order-ing transition is not detected by XRD so far [19], butthe lattice phonons show some changes [15]. A largechange of electronic structure which acquires 1 D char-acter [19, 22] can be a source of this change.An absence of an observable change in magnetic sus-ceptibility can be explained by a coincidence of abso-lute values in the metallic and insulating regimes. Onthe other hand, a similar absence of a change in mag-netic susceptibility on charge localization in TMTTF -based materials [33] was suggested as an evidence ofcharge and spin degrees of freedom being decoupled [34].These materials show 1D electronic behavior, and arewell-described by the Bonner-Fisher (BF) model of mag-netic susceptibility. Typically, they show a transition intoa magnetically ordered or spin-Peierls state at temper-atures about ten times lower than the temperature ofthe metal-insulator transition [33]. There is an apparentqualitative similarity between the decoupling of the spinand charge degrees of freedom in 1D TMTTF salts andthe behavior of charge and spin properties of κ -HgCl,though the difference between the temperatures of theinsulating transition and the decrease of χ M ( T ) is muchsmaller.An evidence of an emergent 1D behavior of the chargedegree of freedom [16, 19, 20] suggests, that a suitablemodel is a BF model for magnetic susceptibility of S=1/2antiferromagnetic 1 D chains [35],[36]. While the tem-perature range of 24 K-30 K is too narrow to extractvalues of magnetic exchange J from a fit to the experi-mental data, we can still compare the relevant tempera-ture dependence. Fig 2 (a) green curve shows susceptibil-ity calculated by the numerical approximation of the BFmodel [37] with one electron per (BEDT-TTF) dimerand J =400 K. The decrease of χ M ( T ) on cooling belowT=24 K is much faster than the BF or triangular latticemagnetic susceptibility [4], and cannot be understoodwithout considering AF or spin singlet correlations or or-dering. The decrease of susceptibility was also observedin ESR measurements [21]. Decreasing susceptibility ina small temperature range between 24 K and 15 K canbe fit by a magnetic susceptibility of a gapped system e − ∆ kBT , which yields a gap ∆ =36 K=1.5 Tc. A transitionto a spin-singlet or anitiferromagnetically ordered statecan be expected in this charge ordered system, basedboth on theory [25] and on experimentally observed be-havior of other charge-ordered materials [26, 33].Heat capacity in the region of 24 K does not showany phase transition features, but this measurementsmethod is known to be not to be a sensitive indicator ofa magnetic transition neither in 1 D [38], nor in 2 D [39]organic-based systems.Due to its high sensitivity, the most reliable informa-tion is obtained by the cantilever torque measurements.On a spin singlet transition an amplitude of magnetictorque should decrease down to zero, however magnetictorque of κ -HgCl preserves its paramagnetic characterbelow 24 K, and does not detect any signatures of mag-netic ordering. A weak decrease of torque amplitude τ θ in ab plane, and of phase values in ac plane observed at1 T (Fig. 2) follow the susceptibility behavior. As awhole, this decrease of susceptibility without long rangeorder detected suggests that AF order or spin singletpairs might be formed with a certain correlation lengthin the temperature range between 24 and 15 K.Below 15 K magnetic susceptibility starts to increaseagain. This change occurs when κ -HgCl enters the tem-perature regime where the re-entrant behavior resultingin melting of the charge order is observed by Raman scat-tering [16]. The increase of χ M ( T ) below 15 K cannotbe fully described as a response of magnetic impuritiesproducing the Curie-like behavior: It shows weaker tem-perature dependence and flattens below 5 K, instead ofdiverging as 1/T. In the temperature regime where the charge order ismelting, magnetic susceptibility will present a superpo-sition of the susceptibility of charge-ordered static orslowly fluctuating ( χ CO ) and non-ordered ( χ H ) com-ponents of the system, as well as susceptibility of do-main walls between ordered and non-ordered fractions χ D : χ M = ρχ H +(1 − ρ ) χ CO + χ D . Here ρ is a part of thesystem where the charge order is melted. It is controlledby temperature, increasing from ρ = 0 in the charge or-dered system above 15 K, to ρ = 1 / ρ .Basing on the behavior of susceptibility in the 24-15 K range, we can assume χ CO =0. The susceptibil-ity of the non-charge-ordered fraction of the system χ H would depend on the dimensionality; BF susceptibilityin the Fig. 2(a) provides some estimate for non-orderedsusceptibility for ρ = 1. The increase of χ M ( T ) on cool-ing between 15 and 5 K can be associated an increaseof ρ on cooling, but would also include the response ofthe domain walls between ordered and non-ordered do-mains. It is known, that non-interacting free spins atthe ends of charge ordered chains can produce a Curielaw dependence in χ D ( T ) [37], while interactions be-tween domain walls lead to a more complicated picture(Ref. [40] and references therein), which can be relatedto the magnetic susceptibility of κ -HgCl. Two spinson the ends of fluctuating charge order chains inter-acting through a non-charge-ordered dimer domain wallare suggested to produce dynamic ferromagnetic inter-actions. The relevant response observed in an increaseof χ M ( T ) on cooling for the dipole liquid candidate κ -(BEDT-TTF) Hg(SCN) Br [41]. It worth noting, thatthe increase and saturation of χ M ( T ) below 15 K in κ -HgCl is similar to this response, but on a much smallerscale. Ref. 40 demonstrates that orphan spins of domainwalls produce a paramagnetic behavior in torque, withtemperature and magnetic field dependence of the phase θ and amplitude τ θ of the paramagnetic component. In κ -HgCl torque data show this effect. It is weak, and isonly observed at 1 T, suggesting that the impurity spinsare saturated at higher fields.Overall the data suggest that the increase of the χ M ( T )on cooling and the temperature dependence of torqueat low fields is produced by interacting domain walls,which are fully polarized already at 3 T according tothe torque measurements, and the torque signal in κ -HgCl is dominated by the bulk response of paramagneticspins. This result is in agreement with a suppression ofa peak in T − in NMR [42] response by the magneticfield directed perpendicular to BEDT-TTF-based layers,and confirms that the peak is due to the response of thedomain walls. According to Raman data in magneticfield in this direction, the charge distribution itself doesnot change in field up to 30 T [43].No indication of magnetic ordering is observed in mag-netic cantilever torque studies of κ -HgCl down to 120mK, with paramagnetic torque amplitude τ θ ∝ H upto the highest measured field of 17.5 T, and the phasebeing constant with field. This absence of magnetic or-der or of a spin singlet state in an insulator κ -HgCl withone unpaired electron (S=1/2) and estimated magneticexchange J of the order of 100 K or higher [15, 22, 41]is suggestive of a spin liquid state, the same conclusionthat came from NMR measurements [42]. However, thelow temperature magnetic state in κ -HgCl has signaturesvery different from spin liquid candidates triangular 2Dorganic Mott insulators.The first essential difference between κ -HgCl and othertriangular lattice Mott insulators is the spectrum of mag-netic excitations. In S=1/2 triangular lattice organicMott insulators without a detected charge degree offreedom, such as κ -(BEDT-TTF) Cu (CN) [15, 44], κ -(BEDT-TTF) Ag (CN) [45], and Pd(dmit) -based ma-terials [46] Raman scattering spectroscopy detects a con-tinuum of magnetic excitations that corresponds to the-oretically suggested excitation spectrum of a triangularlattice with S=1/2 [11, 47–49]. This continuum is ab-sent in κ -HgCl [15]. While magnetic excitations is thismaterial are still to be detected, the difference indicates adifferent magnetic state, possibly due to 1D properties of κ -HgCl below 30 K, or magnetic interactions decreasedby more than one order of magnitude.In contrast to 2D triangular spin liquid candidates [6,50], κ -HgCl shows γ =0 linear term in the heat capac-ity [15]. This is an expected result for a non-orderedS=1/2 antiferromagnetic chain [35, 51], and is anotherevidence of 1D behavior.Below 15 K the system looses signatures of short rangemagnetic correlations, with charge fluctuations and re-sponse of domain walls becoming dominant. Basing ona relatively weak response on the domain walls, we canargue that their concentration is pretty small. As appar-ent from the data, and do not lead to a possible randomsinglet state that can be induced by disorder [52, 53].This suggests that a scenario of 1D S=1/2 chains wherefluctuating charge degree of freedom is coupled to spinsand prevents their order is the most relevant. ACKNOWLEDGEMENTS
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Single crystals of κ -(BEDT-TTF) Hg(SCN) Cl ( κ -HgCl) were prepared byelectrochemical oxidation of the BEDT-TTF solu-tion in 1,1,2-trichloroethane (TCE) at a temperatureof 40 ◦ C and a constant current of 0.5 µ A. A solutionof Hg(SCN) , [Me N]SCN · KCl, and dibenzo-18-crown-6in 1:0.7:1 molar ratio in ethanol/TCE was used assupporting electrolyte for the κ -HgCl preparation. Thecomposition of the crystals was verified by electronprobe microanalysis and X-ray diffraction. Heat capacity
Heat capacity was measured usingQuantum Design PPMS system equipped with the DRoption for crystals of the mass 2-4 mg.
Cp/T ( JK-2mol-1 ) T ( K ) FIG. 5. Heat capacity of κ -(BEDT-TTF) Hg(SCN) Cl in therange 100 mK - 3 K. Note γ = 0 within the error of the mea-surements. Magnetic susceptibility
Bonner-Fisher susceptibility (Fig. 2(a), green curve)was calculated following the numerical approximationform Ref. [37] χ BF = N g µ B k B T ( 0 .
25 + 0 . x + 0 . x . x + 0 . x + 6 . x ) , (2)where x = | J | /k B T with one electron per (BEDT-TTF) dimer and J=400 K. c m(emu/mol) T e m p e r a t u r e ( K )
FIG. 6. Magnetic susceptibility of κ -(BEDT-TTF) Hg(SCN) Cl polycrystal sample measured by SQUIDmagnetometer using H=1 T in the temperature range from300 to 2 K.
Fig. 6 shows magnetic susceptibility data in the wholemeasured range. The figure also demonstrated a curve of χ dimer = Ae − ∆ kBT which can describe χ M ( T ) in the tem-perature range between 24 and 15 K, and χ Curie =C/Tto compare to the low-temperature behavior of χ M ( T ). Cantilever magnetic torque measurements
Fig 7 presents an example of analysis of the can-tilever magnetic torque data. The full torque τ (blackcurve) is reproduced well by the sum of τ θ sin( θ − θ ),and τ θ sin2( θ − θ ) components. Calibration of the can-tilever response by gravity signal is discussed in detailsin Ref. [50] -100 0 100-0.000050.000000.00005 I n t en s i t y ( a . u . ) q (deg) raw data t q sin( q-q ) torque component fitting curve t q sin( q-q ) t sin2( q-q ) fitting curve t sin2( q-q ) (ab) plane 5 T 5 K -100 0 100-0.00010-0.000050.000000.00005 M agne t i c t o r que t ( a . u . ) q (deg) 5 K 10 K 15 K 20 K 25 K 29 K 34 K5T FIG. 7. Upper panel: Cantilever magnetic torque data anal-ysis, and example for the data for the rotation in ab planeat 5 K in magnetic field of 5 T. The figure shows originaldata, and τ θ sin( θ − θ ), and τ θ sin2( θ − θ ) components withthe respective fitting curves. Lower panel: Raw data of κ -(BEDT-TTF) Hg(SCN) Cl obtained by cantilever magnetictorque measurements. Measurements are done for rotationin ab plane, H= 5T. Angle dependence of τ θθ