Magnetic Torque due to Anisotropic Diamagnetism in Neutral BEDT-TTF Crystals

Shinya Uji^1, , Taichi Terashima¹, Shiori Sugiura², Satoshi Iguchi², Takahiko Sasaki², Kenichiro Hashimoto³, Masaaki Shimozawa⁴, Naoki Yoneyama⁵, and Reizo Kato⁶

In the last two decades, fascinating magnetic states, so-called quantum spin liquids (QSLs), have been found in some organic materials. Conventional spin systems with antiferromagnetic exchange interaction show long range orders at low temperatures, where the spin entropy vanishes. However, spins in QSLs are strongly entangled and remain in highly fluctuating states accompanied with high entropy even at very low temperatures. So far, three series of organic QSLs have been reported; κ-(BEDT-TTF)₂\(M_{2}\)(CN)₃ (M = Cu, Ag),^1–6) EtMe₃Sb[Pd(dmit)₂]₂,^7–10) and κ-H₃(Cat-EDT-TTF)₂.¹¹⁾ Their significant common properties are a temperature-independent spin susceptibility \(\chi_{0}\) and a \(\gamma T\) term in the heat capacity at very low temperatures, suggesting gapless spin excitations accompanied by the formation of spinon Fermi surface. In the above organic QSLs as well as other spin systems, micro-cantilevers^12,13) have been often used to detect the magnetic torque signal^4,10,11) in wide field and temperature ranges because of very high sensitivity.

The magnetic torque is given by \({\boldsymbol{\tau}} = -\partial F/\partial\theta\), where \(F = -\int_{0}^{H}\mu_{0} M(\theta)\,dH\) is the magnetic free energy and θ is the field angle from a major crystal axis. The anisotropic magnetization M is given by \begin{equation} M(\theta) = M_{\|}\sin^{2}(\theta - \theta_{0}) + M_{\bot}\cos^{2}(\theta - \theta_{0}), \end{equation} (1) where \(M_{\|}\) and \(M_{\bot}\) are the principal values of the magnetization in a crystal plane. Assuming the magnetization linear to the field, \(M_{\|} =\chi_{\|}H\) and \(M_{\bot} =\chi_{\bot} H\), we obtain the magnetic torque, \begin{align} {\boldsymbol{\tau}} &= - \frac{\partial F}{\partial \theta} = \frac{1}{2}\mu_{0}(\chi_{\|} - \chi_{\bot})H^{2}\sin[2(\theta - \theta_{0})] \notag\\ &\propto \Delta \chi H^{2}\sin[2(\theta - \theta_{0})], \end{align} (2) where \(\Delta\chi =\chi_{\|} -\chi_{\bot}\) is the anisotropic part of the susceptibility. Here we should note that the torque signal is proportional to \(\Delta\chi\) (not the absolute value of the susceptibility) and vanishes in an isotropic case (\(\Delta\chi = 0\)).

In the QSLs, the constant spin susceptibility \(\chi_{0}\) is observed at low temperatures, which is comparable to Langevin diamagnetism arising from the closed shells of all the atoms. The diamagnetism is given by \(\chi_{\text{dia}}\propto - Ze^{2}\langle r^{2}\rangle\), where Z is the atomic number and \(\langle r^{2}\rangle\) is the mean square distance of the electrons from the nucleus. Because of the closed shell properties, the diamagnetism is expected to be temperature independent and isotropic, yielding no torque signal. Therefore, the constant torque signal observed at low temperatures in the QSLs has been recognized as consistent evidence of the \(\chi_{0}\) term.^4,10,11) In this study, we find that the diamagnetism of a typical neutral organic molecule BEDT-TTF with no local spins is anisotropic due to the planar structure and leads to significant torque signals. The results clearly show that the toque signal can arise from the anisotropic diamagnetic susceptibility in addition to the anisotropic spin susceptibility in any organic molecular QSLs; careful consideration is required in the quantitative discussion of the torque signals in various molecular materials including organic QSLs.

The BEDT-TTF molecule is composed of C, S, and H atoms as shown in Fig. 1(a). The crystal structure of the neutral BEDT-TTF crystals is monoclinic \(P2_{1}/c\) with \(a = 0.6642\) nm, \(b = 1.402\) nm, \(c = 1.577\) nm, and \(\beta = 93.83\)°.¹⁴⁾ The magnetic susceptibility is measured by a Quantum Design MPMS SQUID magnetometer. The magnetic torque is measured by a piezo-micro-cantilever.^11,12) A single crystal is cut into two pieces and then each crystal (∼25 µg) is attached to a different cantilever with silicone grease as shown in Fig. 1(b). The two crystals are rotated in two different planes; H in \((0\bar{1}1)\) and (100) planes as depicted in Figs. 1(c) and 1(d), respectively. The signal is detected by a lock-in amplifier with a frequency of ∼15 Hz, using a homemade bridge circuit.¹³⁾ The background signal arising from the cantilever bending by the sample gravity, which has a \(\sin(\theta)\) dependence, is numerically subtracted.

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Figure 1. (Color online) (a) Structure of BEDT-TTF [bis(ethylenedithio)tetrathiafulvalene] molecule. (b) Photos of microcantilevers with BEDT-TTF single crystals. Unit cells projected on (c) \((0\bar{1}1)\) and (d) (100) planes. Here, \(\boldsymbol{a}^{*}\) is defined as the axis perpendicular to the \(\boldsymbol{b}\) and \(\boldsymbol{c}\) axes.

Figures 2(a) and 2(b) present typical torque curves as a function of the field angle θ for sample #1 [H in the \((0\bar{1}1)\) plane] and #2 [H in the (100) plane], respectively, whose field angles are defined in Figs. 1(c) and 1(d). In the whole temperature region, sinusoidal curves are observed for both samples. The torque curves can be fitted with a functional form, \(\tau (\theta) =\tau_{0}\sin[2(\theta -\theta_{0})]\), where \(\tau_{0}\) and \(\theta_{0}\) are the amplitude and phase of the two-fold symmetry term, respectively. As shown in Fig. 3(a), the amplitude \(\tau_{0}\) is much larger for #1 than for #2, showing larger anisotropy of the diamagnetism in the \((0\bar{1}1)\) plane. The crystal shape effect, the anisotropic demagnetization factor has a negligible contribution to the torque signal. We see a broad minimum at ∼180 K for #1 in contrast to a monotonic increase with decreasing temperature for #2. The anisotropy ratio, \(\tau_{0}(\#1)/\tau_{0}(\#2)\) monotonically decreases with decreasing temperature. This temperature dependence mainly arises from that of \(\tau_{0}(\#2)\) for H in (100) planes, whose origin is not clear. The thermal compression with temperature, which leads to some deformation of the molecular orbital, may be responsible for the temperature dependence.

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Figure 2. (Color online) Torque curves as a function of the field angle θ for (a) sample #1 [H in the \((0\bar{1}1)\) plane] and (b) #2 [H in the (100) plane]. The angles are defined in Fig. 1. Each curve is shifted for clarity.

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Figure 3. (Color online) Temperature dependence of (a) \(\tau_{0}\) and (b) \(\theta_{0}\) for samples #1 and #2, obtained by fitting the torque curves with \(\tau (\theta) =\tau_{0}\sin[2(\theta -\theta_{0})]\).

The phase \(\theta_{0}\), as shown in Fig. 3(b), is almost independent of temperature within the experimental error for both samples. The torque as a function of \(H^{2}\) is presented in Fig. 4. For both samples, the quadratic dependence \(\tau\propto H^{2}\), showing \(M\propto H\), is evident at 1.7 and 50 K. No saturation behavior at 1.7 K in high fields clearly shows that the torque signal arises predominantly from the diamagnetism but not from impurity spins.

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Figure 4. (Color online) Torque signals as a function of \((\mu_{0}H)^{2}\) at 1.7 and 50 K for samples #1 and #2.

The magnetization measured by the SQUID magnetometer is plotted as a function of temperature for two field directions in Fig. 5(a). The susceptibility is larger for \(H\parallel [011]\) than for \(H\parallel [100]\), and both values gradually decrease with decreasing temperature. The difference between the susceptibilities for \(H\parallel [011]\) and \(H\parallel [100]\) is almost independent of temperature, which is consistent with \(\tau_{0}(\#1)\) in Fig. 3(a) within the experimental error.

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Figure 5. (Color online) (a) Temperature dependence of magnetic susceptibility at 5 T. Dotted (red) curve and dashed (green) line indicate a Curie law for \(S=1/2\) with a concentration of 2% and the Langevin diamagnetism estimated from the Pascal constants, respectively. (b) Magnetization measured at 2 K as a function of field.

An upturn below 20 K for \(H\parallel [011]\) may be due to impurity spins, which is approximately given by a Cuire law for \(S = 1/2\) with a concentration of 2% (dotted curve). Using the Pascal constants of C, S, and H, we obtain the Langevin diamagnetism, \(-2.04\times 10^{-4}\) emu/mol (dashed line), which is comparable to the experimental values. The field dependence of the magnetization (M–H curve) is presented in Fig. 5(b). The magnetization is proportional to the field, consistent with the torque results (Fig. 4). In the M–H curve, we observe no appreciable contribution from the impurity spins.

The Langevin diamagnetism leads to an NMR shift of the nuclei, a chemical shift. The chemical shift depends on the electron density surrounding the atom, whose orbitals are polarized by the external magnetic field. Since the molecular orbitals are anisotropic, reflecting the planar structure of the molecule, the chemical shift could be anisotropic. Actually, from NMR studies of the central ¹³C nuclei of the BEDT-TTF molecule in various organic conductors,¹⁵⁾ the principal values of the ¹³C chemical shift \(\sigma_{ii}\) for the neutral BEDT-TTF molecule are determined; \(\sigma_{xx} = 46\) ppm, \(\sigma_{yy} = 159\) ppm, and \(\sigma_{zz} = 62\) ppm, where x, y, and z directions are indicated in Fig. 1(a). Although the diamagnetism of the molecule arises from all the atoms, it will be reasonable to assume that the anisotropy of the diamagnetism reflects the chemical shift of the central C atoms since a large portion of the molecular orbital lies on the central C atoms.¹⁶⁾ From the above \(\sigma_{ii}\) (\(i = x,y,z\)) values, we can estimate the anisotropy of the diamagnetism \(M_{\text{dia}}\) of the neutral BEDT-TTF crystal as shown in Fig. 6. Here we should note that the isotropic part of the diamagnetism is unknown, which will be larger than the anisotropic part. We can estimate the torque curves by differentiating \(M_{\text{dia}}\) with respect to θ, which should be compared with the results in Fig. 2. For H in \((0\bar{1}1)\) and (100), we obtain \(\theta_{0} = -7\) and 41°, respectively. These values are roughly consistent with the experimental results, \(\theta_{0}\approx -10\) and 58° in Fig. 3. The calculated anisotropy ratio \(\tau_{0}(\#1)/\tau_{0}(\#2) = 14\) is comparable to the experimental values, 23.5 at 237 K and 10.6 at 52 K. From the absolute amplitude of the torque signal in Fig. 2(a), we obtain the anisotropic part of the diamagnetic susceptibility, \(\Delta\chi \approx 0.8\times 10^{-4}\) emu/mol at 50 K. This value agrees reasonably with the difference between the two susceptibility curves in Fig. 5(a). In this way, we have consistently obtained the anisotropic diamagnetism of the neutral BEDT-TTF crystal in the torque and SQUID measurements.

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Figure 6. (Color online) Calculated magnetization and torque from the chemical shift anisotropy of the central C nuclei. The torque amplitude \(\tau_{0}\) and phase \(\theta_{0}\) are indicated for each torque curve.

Finally we briefly mention the previous torque data in the typical QSL, κ-(BEDT-TTF)₂Cu₂(CN)₃.⁴⁾ At 10 T, the torque has a broad maximum at ∼30 K and decreases with decreasing temperature. Below 3 K, the torque becomes constant, \(\tau_{0}\approx 10^{5}\) dyn cm/mol above 10 T. Assuming the valence of the molecule (BEDT-TTF)^+0.5, we can estimate the chemical shifts, \(\sigma_{xx} = 112\) ppm, \(\sigma_{yy} = 175\) ppm, and \(\sigma_{zz} = 56\) ppm of the central C nuclei.¹⁵⁾ This chemical shift anisotropy will give the diamagnetic torque \(\tau_{0}\approx 3\times 10^{4}\) dyn cm/mol in κ-(BEDT-TTF)₂Cu₂(CN)₃. This value amounts to a large fraction, ∼30% of the total torque signal.

In conclusion, the diamagnetic susceptibility of the neutral BEDT-TTF crystals is anisotropic, reflecting the planar structure of the molecule. The anisotropic diamagnetism leads to significant torque signals, which are slightly temperature dependent. The anisotropy of the torque signal is roughly consistent with the chemical shift of the central C nuclei of the BEDT-TTF molecule. The presence of the anisotropic diamagnetism shows that the torque signal should be carefully discussed in various organic materials, especially organic QSLs.

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