Monomer Mott Insulator (BEDT-TTF)Cu[N(CN)₂]₂ as a Potential Nodal Line System

Naoki Yoneyama^1, , Muhammad Khalish Nuryadin² , Takao Tsumuraya³ , Satoshi Iguchi² , Takahiro Takei⁴ , Nobuhiro Kumada⁴ , Masanori Nagao⁴ , Isao Tanaka⁴ , and Takahiko Sasaki²

Electronic band structures with linear dispersion have recently attracted attention because of the massless Dirac nature of the fermion carriers. The Dirac semimetal (DS) state can be achieved when the Fermi level is located near the center of the dispersion. The discovery of the quantum Hall effect in graphene¹⁾ as a two-dimensional (2D) DS with a cone-type dispersion has drawn considerable attention. In particular, organic molecule-based compounds are a suitable platform for investigating the DS state of a bulk system. The first DS state in organics was identified in the 2D layered system of α-(BEDT-TTF)₂I₃²⁾ and the studies on the relative compounds are recently developing,^3,4) where BEDT-TTF is bis(ethylenedithio)tetrathiafulvalene (abbreviated below as ET). The electronic band near the Fermi level in these systems is derived from the HOMO orbital of the ET molecule. In contrast, in the single-component conductor [Pd(dddt)₂], the DS state with nodal lines near the Fermi level originates from the HOMO and LUMO multibands, that arises under pressure.⁵⁾ Moreover, [Pt(dmdt)₂] realizes the node-line DS state at ambient pressure.⁶⁾ The three-dimensional (3D) diamond lattice system of (ET)Ag₄(CN)₅ with a half-filled band^7,8) is another interesting candidate for DS.

Here, we focus on an ET salt with a Cu(I) dicyanamide counter anion, (ET)Cu[N(CN)₂]₂,⁹⁾ which is classified as a modified analog of (ET)Ag₄(CN)₅. (ET)Cu[N(CN)₂]₂ is a byproduct of the organic dimer Mott insulator/superconductor κ-(ET)₂Cu[N(CN)₂]Cl (\(T_{c}=12.8\) K, under 0.3 kbar),⁹⁾ one of the most notable 2D-layered conducting systems comprising strongly dimerized ET molecules. In contrast to the dimer Mott system, (ET)Cu[N(CN)₂]₂ can be considered as a spin-1/2 ET\(^{\bullet +}\) monomer Mott insulator. There is no 2D conducting sheet structure in (ET)Cu[N(CN)₂]₂, in which a peculiar 3D anisotropic diamond-like network of ET molecules has been overlooked for decades.¹⁰⁾

In this study, we investigated the band structure at room temperature (RT) and observed that the uniform zigzag chain with dihedral interchain (anisotropic diamond-like) interaction resulted in a potential DS state with nodal lines. Using X-ray crystal structure analysis, dc resistivity, and static magnetic susceptibility measurements, the paramagnetic insulating properties were interpreted in terms of the monomer Mott insulating state.

Single crystals of (ET)Cu[N(CN)₂]₂ were grown using conventional electrochemical methods.⁹⁾ The black needle-like crystalline shape with the typical dimensions of \(2\times 0.05\times 0.02\) mm³ can be easily distinguished visually from the thick platelet byproduct κ-(ET)₂Cu[N(CN)₂]Cl. We performed X-ray structural analyses at 296 and 100 K with single crystals (Rigaku, XtaLAB mini and VariMax DW). For band calculations, we utilized crystallographic data from the literature,⁹⁾ where the coordinates of the H and N atoms were optimized for density functional theory (DFT) calculations. In the tight-binding model, the intermolecular overlaps between the HOMO orbitals of ET are calculated using the extended Hückel method.¹¹⁾ We employed projected augmented-wave pseudopotentials in the DFT band calculations^12,13) with plane-wave basis sets as implemented in Quantum ESPRESSO.¹⁴⁾ The cut-off energies for the plane waves and charge densities were set to 45 and 488 Ry, respectively. The exchange–correlation functional is a generalized gradient approximation proposed by Perdew, Burke, and Ernzerhof.¹⁵⁾ The dimensions of the \(\boldsymbol{k}\)-point mesh were \(12\times 12\times 12\). The dc resistivity was measured at ambient pressure using the conventional two- or four-terminal methods. The quasi-four-terminal method¹⁶⁾ with a constant current of 0.1 µA was adopted for measurements under pressure. A CuBe clamp-type hydrostatic pressure cell was used for the sample in the \(E\parallel c\) configuration up to 1.8 GPa with Daphne 7373 oil. The magnitude of the pressure labeled below was estimated from the load gauge of the press machine at RT. Pressure calibration performed distinctively using the superconducting transition of Pb exhibited a pressure loss of approximately 0.3 GPa (at ∼10 K) until 2 GPa. The temperature was controlled using a PPMS system (Quantum Design, Dynacool). Temperature dependence of static magnetic susceptibility was measured using a SQUID magnetometer (Quantum Design, MPMS-XL), using single crystals with a total weight of 1.1 mg. A batch of samples was placed inside a polyacetal rod that functioned as a sample holder. The samples were inserted into the center of the rod without grease and the long axis of the multiple crystals was positioned perpendicular to the applied field (\(H\perp c\)). We obtained the magnetic susceptibility (\(\chi_{\text{dc}}\)) after subtracting the diamagnetic contribution using Pascal's constant, \(\chi_{\text{dia}}=-3.6\times 10^{-4}\) emu/mol, from the measured data.

First, we describe the crystal structure. The monoclinic (\(C2/c\)) crystal structure reported previously⁹⁾ was well-reproduced at 296 K and no significant variation was observed at 100 K.¹⁷⁾ One-half crystallographically independent ET molecule exists in the crystal structure of (ET)Cu[N(CN)₂]₂. The formal charge of ET is +1 owing to the total valence of the anion (Cu(I)[N(CN)₂]₂)⁻¹, which is consistent with the charge estimated from the intramolecular bond length of ET at both 100 and 296 K. Figure 1(a) shows the crystal structure viewed from the molecular long axis of ET. The ET molecules form one-dimensional (1D) zigzag chains at regular intervals along the c-axis with the nearest intermolecular (diagonally side-by-side) S-S contact (bold line, t). There is a two-fold rotation axis perpendicular to the molecular plane, which guarantees the uniformity of the 1D chain. In addition to intrachain coupling, the second-nearest interaction (wedge symbol, \(t'\)) connects an ET with two others on neighboring chains. Therefore, all the ET molecules possess distorted tetrahedral coordination geometry. A schematic of the 3D network is shown in Fig. 1(b). The spheres (centroids of ET) construct a uniform 1D zigzag chain along the c-axis, which is surrounded by four other chains via the \(t'\) couplings. This corresponds to a strongly anisotropic (almost 1D) diamond-like structure.

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Figure 1. (Color online) Crystal structure of (BEDT-TTF)Cu[N(CN)₂]₂. (a) View from the ET molecular long-axis showing the uniform 1D zigzag chains along the c-axis. The colored 1D chain through the middle of the unit cell is located on the bc plane. The bold and pale-colored ET motifs are at the front and rear sides, respectively. The anions have been omitted for simplicity. (b) Diamond-like lattice corresponding to the ET stacking shown in (a). (c) View from the b-axis, showing the polymeric anion chain along the \(a+c\) direction in color. (d) View along the anion chain shown in (c).

The polymeric anion was composed of tetrahedral coordinates. The Cu(I) atoms were bridged by two bent dicyanamide anions [Fig. 1(c)] along the \(a+c\) direction. Owing to the closed-shell structure expected for the anion, it is unlikely to contribute to the electronic state near the Fermi level and properties such as conductivity and paramagnetism. In Fig. 1(d), the view from the \(a+c\) direction indicates that the diamond-like packing of ET contains a columnar cavity occupied by polymeric anions, which results in an unusual donor/anion mixed-stacking structure.

Next, we describe the band structure calculated using the tight-binding model. The transfer integrals t and \(t'\) were estimated as −0.192 and +0.0229 eV, respectively. Almost the same values for t (\(=-0.193\) eV) and \(t'\) (\(=+0.0227\) eV) were obtained using our X-ray data at 100 K, indicating that the band picture presented below is safely preserved to at least 100 K. The conventional (monoclinic C) unit cell was reduced to a primitive cell with lattice vectors \(\boldsymbol{a}'=(\boldsymbol{a}-\boldsymbol{b})/2\), \(\boldsymbol{b}'=(\boldsymbol{a}+\boldsymbol{b})/2\), and \(\boldsymbol{c}'=\boldsymbol{c}\), as shown in Fig. 2(a). The primitive cell constants were \(a'=b'=10.886\) Å, \(\alpha'=\beta'=115.45\)°, and \(\gamma'=79.63\)° (\(V'=V/2=993.2\) ų and \(Z'=Z/2=2\)). The corresponding first Brillouin zone (BZ) is shown in Fig. 2(b) and the primitive cell is used for the band calculations.

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Figure 2. (Color online) (a) Primitive unit cell \((a', b')\) projected on the (001) plane of the conventional monoclinic C cell. (b) First Brillouin zone for the primitive cell with two nodal lines (i) and (ii). The solid and broken curves denote the inside and outside of the BZ, respectively. (c) Band structure and density of states (DOS) calculated within the tight-binding approximation.

Based on the standard tight-binding approximation, the matrix elements of the \(2\times 2\) secular equation are \(H_{\text{AA}}=H_{\text{BB}}=0\) and \(H_{\text{AB}}=H^{*}_{\text{BA}}=t+te^{-ik_{c'}c'}+t'e^{ik_{a'}a'}+t'e^{-ik_{b'}b'-ik_{c'}c'}\). The energy dispersion is then obtained as: \begin{equation} E_{\pm}(\boldsymbol{k}) = \pm 2\sqrt{(t\alpha_{k}+t'\beta_{k})^{2}+2tt'\alpha_{k}\beta_{k}[\cos(a'k_{a'}-b'k_{b'})-1]}, \end{equation} (1) where \(\alpha_{k}=\cos(k_{c'}c'/2)\) and \(\beta_{k}=\cos(k_{a'}a'/2+k_{b'}b'/2+k_{c'}c'/2)\). There are two band dispersions corresponding to the two equivalent ET molecules. Considering the transfer of one electron from the HOMO of ET to the charge compensating anions, the Fermi level is located in the middle of the HOMO band, which results in a half-filled band. The empty upper (\(E_{+}\)) and fully filled lower (\(E_{-}\)) bands are symmetric with respect to the Fermi level, and they coincide at Z \((k_{a'},k_{b'},k_{c'})=(0,0,\pi/c')\) as shown in Fig. 2(c). This band degeneracy at Z originates from the zigzag uniform-chain structure. Neglecting a small \(t'\) (\(\ll t\)) Eq. (1^,) reduce to \(E_{\pm}(\boldsymbol{k})\approx \pm 2|t\cos(k_{c'}c'/2)|\), which leads to 1D “nodal” Fermi surfaces \((0 0 (\pm\pi/c'))\) on Z with a linear energy dispersion around the 1st BZ. In the present system, a weak \(t'\) breaks the 1D Fermi degeneracy and contributes to lead to nodal lines that orthogonally cross at Z as described below. The two nodal lines (i) and (ii) were obtained from Eq. (1^,) using the conditions \(E_{\pm}(\boldsymbol{k})=0\):¹⁸⁾ (i) \(k_{c'}c'=\pi\) and \(k_{a'}a'+k_{b'}b'=0\), and (ii) \(t\cos(k_{c'}c'/2)+t'\cos(k_{a'}a'+k_{c'}c'/2)=0\) and \(k_{a'}a'-k_{b'}b'=0\). The nodal lines are shown schematically in Fig. 2(b). Node (i) is linear along the \(b^{*}\) (\(=-a'^{*}+b'^{*}\))-axis on the 1st BZ, whereas node (ii) bends along the \(a^{*}\) (\(=a'^{*}+b'^{*}\)) direction on the \((1\bar{1}0)\) plane, namely, the \(c^{*}a^{*}\)-plane in the monoclinic cell. Therefore, strictly speaking, the present nodal feature does not originate from the diamond-like structure (\(t'\)), but from the zigzag uniform stacking along the c-axis (t) corresponding to the existence of two equivalent sites in a primitive unit cell. Thus, a slight modification to the uniform zigzag chain, such as the perturbative dimerization of ET molecules, can easily cause gap formation on all nodal lines.

Our tight-binding model, based on the extended Hückel method, generally reproduces the DFT band structure shown in Fig. 3. Although the bands derived from the anion orbitals overlapped with the lower HOMO band, they did not contribute to the formation of the Dirac-type dispersion.

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Figure 3. Band structure of (BEDT-TTF)Cu[N(CN)₂]₂ obtained via first-principles DFT calculations. The Fermi level is set to zero energy at the Dirac point Z (dotted line).

We now describe the experimental results. Figure 4 shows the Arrhenius plot of the dc resistivity with the electric field parallel to three directions (\(a^{*}\), b, and c-axes). The c-axis (crystal long axis) is the most conductive direction, with a resistivity of approximately 20 Ω cm at RT. The resistivities along the b and \(a^{*}\)-axes are four and five orders of magnitude higher than that along the c-axis, respectively; \(\rho_{c}\ll \rho_{b}\leq \rho_{a^{*}}\). The high anisotropy is consistent with the quasi-1D band dispersion. The behavior of the resistivity in the entire temperature range measured below ≈200 K can be explained as an insulator with an excitation energy \(E_{a}\) of approximately 0.13 eV (solid line in \(E\parallel c\)), which is comparable to the results of a previous study.⁹⁾ A slight nonlinearity in the Arrhenius plot was observed above 200 K, whose high reproducibility was confirmed using several specimens. The deviation above 200 K will be related to non-ohmic behavior [current-dependent \(\rho(T)\)] observed recently.¹⁰⁾

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Figure 4. (Color online) Arrhenius plot of the dc resistivity of (BEDT-TTF)Cu[N(CN)₂]₂. The anisotropic resistivities along the three directions are measured under ambient pressure; the resistivity under pressure is measured in \(E\parallel c\). The solid lines are obtained by linear fitting below 200 K, indicating the pressure dependence of the activation energy (inset).

The present compound has a half-filled band; thus, the most reasonable interpretation of the insulating behavior is the scenario of Mott insulator owing to the strong on-site Coulomb interaction \(U_{\text{eff}}\). As the localized site should be an ET molecule (monomer), \(U_{\text{eff}}=U_{0}-V\), where \(U_{0}\) is the on-site Coulomb interaction for an ideally isolated ET, and V is the sum of the inter-site interactions. \(2E_{a}\) is equivalent to the Hubbard gap, \(E_{g}=U_{\text{eff}}-W\), where W is the bandwidth. Applying \(E_{a}=0.13\) eV (obtained experimentally) and \(W=4|t-t'|=0.86\) eV (estimated from the tight-binding model), we obtained \(U_{\text{eff}}=2E_{a}+W=1.12\) eV. The magnitude of \(U_{\text{eff}}\) was almost comparable to that of the other monomer Mott insulators: 0.78 and 0.82 eV for (ET)Ag₄(CN)₅¹⁹⁾ and ζ-(ET)PF₆,²⁰⁾ respectively.

In the resistivity measured under pressure along the c-axis, the insulating behavior was still observed up to 1.8 GPa. As shown in the inset of Fig. 4, \(E_{a}\) monotonically decreases with increasing pressure. A rough extrapolation at a rate of \(dE_{a}/dP\approx -20\) meV/GPa (broken line in inset) implies that a pressure significantly higher than 5 GPa is required to completely suppress the charge gap. The present salt appeared to be robust against pressure, compared to (ET)Ag₄(CN)₅, \(dE_{a}/dP\approx -34\) meV/GPa.⁸⁾ Unfortunately, in the latter salt, the inevitable disorder in the anion (C/N site occupancy) disturbed the realization of a DS state under pressure. A notable feature of the present salt is the absence of such disorder in Cu[N(CN)₂]\(_{2}^{-}\). Thus, if the crystal structure is preserved under pressure, then the node line DS state can arise in a weak-limit condition of the electron correlation.

The localized carrier of (ET)Cu[N(CN)₂]₂ has an \(S=1/2\) spin degree of freedom, as observed in ESR.⁹⁾ The strong 1D ET network suggests a low-dimensional feature in the temperature dependence of the paramagnetic susceptibility (\(\chi_{\text{para}}\)). Figure 5 shows \(\chi_{\text{para}}(T)\) at 1 T (closed circles), which is obtained from \(\chi_{\text{dc}}(T)\) (open circles) after subtracting the Curie term of approximately 1.1% per ET molecule [\(\chi_{\text{imp}}(T)\)]. The amplitude of \(\chi_{\text{para}}\), which is approximately \(2.3\times 10^{-4}\) emu/mol at 300 K, monotonically decreases with temperature. Below ∼25 K, a steep exponential reduction in \(\chi_{\text{para}}\) is observed, indicating a spin-singlet ground state. The inset of Fig. 5 shows \(\chi_{\text{para}}(T)\) at low temperatures with an Arrhenius-type fitting of \(\exp(-\Delta/T)\) using \(\Delta=90\) K.

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Figure 5. (Color online) Temperature dependence of the magnetic susceptibility of (BEDT-TTF)Cu[N(CN)₂]₂. \(\chi_{\text{para}}\) (closed circles) is obtained after subtracting \(\chi_{\text{imp}}\) (broken curve) from \(\chi_{\text{dc}}\) (open circles). \(\chi_{\text{para}}\) is fitted based on the 1D model²²⁾ (solid curve). The inset shows a magnified view of \(\chi_{\text{para}}(T)\) at low temperatures with an Arrhenius fitting (solid curve).

Recently, a ferromagnetic transition at 13 K was reported in wire-shaped (ET)Cu[N(CN)₂]₂.¹⁰⁾ However, the reproducibility of ferromagnetism has not been achieved although we examined several specimens with careful attention to avoid contamination by byproducts. The magnetic ground state is currently an open question; however, our preliminary ESR measurements at low temperatures show the disappearance of the paramagnetic signal below approximately 25 K, which is consistent with the present results for \(\chi_{\text{para}}(T)\) and the spin-singlet state at low temperatures. The ESR results will be reported elsewhere.

Finally, we discuss the paramagnetic behavior. The gradual decrease in \(\chi_{\text{para}}\) from 300 to 25 K is explained as a typical short-range ordering that appears in lower temperature regime of a broad peak in the susceptibility of low-dimensional localized spin systems. Ignoring the interchain interactions originating from \(t'\), the solid curve in Fig. 5 was calculated based on \(S=1/2\) Heisenberg antiferromagnetic spin model in a 1D lattice (Bonner–Fisher model,^21,22) where the exchange interaction is defined as \(-2J\mathbf{S}_{i}\cdot\mathbf{S}_{j}\)) with intrachain exchange interaction \(J=-500\) K and the Bohr magneton per ET of 1.0 \(\mu_{B}\). As another estimation of J, we set the magnitude of \(2J\) to be equal to the energy gap between the singlet and triplet states in the 1D Hubbard model, \((-U_{\text{eff}}+\sqrt{U_{\text{eff}}^{2}+16t^{2}})/2\). Using the parameters \(t=-0.19\) eV and \(U_{\text{eff}}=1.12\) eV, we obtain \(|J|\approx 0.059\) eV (\(=650\) K), which is consistent with that obtained by fitting to \(\chi_{\text{para}}(T)\). Although the fitting curve for \(\chi_{\text{para}}(T)\) was not satisfactory, this simple 1D model has a reasonable accuracy as a first-order approximation. The fitting may be improved by considering the interchain exchange interaction.

In summary, we investigated band structure, resistivity, and magnetic susceptibility of (ET)Cu[N(CN)₂]₂. The uniform zigzag chain interaction resulted in a potential DS state with nodal lines, which could appear under pressure owing to the suppression of the observed monomer Mott insulating state. The paramagnetic insulating electronic properties in the half-filled band were well explained in terms of the Mott insulators. The sudden reduction in \(\chi_{\text{para}}\) indicated a spin-singlet (non-magnetic) ground state below 25 K.

References

• 1 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005). 10.1038/nature04233 Crossref, Google Scholar
• 2 S. Katayama, A. Kobayashi, and Y. Suzumura, J. Phys. Soc. Jpn. 75, 054705 (2006). 10.1143/JPSJ.75.054705 Link, Google Scholar
• 3 T. Naito, R. Doi, and Y. Suzumura, J. Phys. Soc. Jpn. 89, 023701 (2020). 10.7566/JPSJ.89.023701 Link, Google Scholar
• 4 S. Kitou, T. Tsumuraya, H. Sawahata, F. Ishii, K. Hiraki, T. Nakamura, N. Katayama, and H. Sawa, Phys. Rev. B 103, 035135 (2021). 10.1103/PhysRevB.103.035135 Crossref, Google Scholar
• 5 R. Kato, H. Cui, T. Tsumuraya, T. Miyazaki, and Y. Suzumura, J. Am. Chem. Soc. 139, 1770 (2017). 10.1021/jacs.6b12187 Crossref, Google Scholar
• 6 B. Zhou, S. Ishibashi, T. Ishii, T. Sekine, R. Takehara, K. Miyagawa, K. Kanoda, E. Nishibori, and A. Kobayashi, Chem. Commun. 55, 3327 (2019). 10.1039/C9CC00218A Crossref, Google Scholar
• 7 Y. Shimizu, A. Otsuka, M. Maesato, M. Tsuchiizu, A. Nakao, H. Yamochi, T. Hiramatsu, Y. Yoshida, and G. Saito, Phys. Rev. B 99, 174417 (2019). 10.1103/PhysRevB.99.174417 Crossref, Google Scholar
• 8 A. Kiswandhi, M. Maesato, S. Tomeno, Y. Yoshida, Y. Shimizu, P. Shahi, J. Gouchi, Y. Uwatoko, G. Saito, and H. Kitagawa, Phys. Rev. B 101, 245124 (2020). 10.1103/PhysRevB.101.245124 Crossref, Google Scholar
• 9 H. H. Wang, U. Geiser, J. M. Williams, K. D. Carlson, A. M. Kini, J. M. Mason, J. T. Perry, H. A. Charlier, A. V. S. Crouch, J. E. Heindl, M. W. Lathrop, B. J. Love, D. M. Watkins, and G. A. Yaconi, Chem. Mater. 4, 247 (1992). 10.1021/cm00020a003 Crossref, Google Scholar
• 10 Y. Huang, T. Mitchell, D. C. Yost, Y. Hu, J. B. Benedict, J. C. Grossman, and S. Ren, Nano Lett. 21, 9746 (2021). 10.1021/acs.nanolett.1c03663 Crossref, Google Scholar
• 11 T. Mori, A. Kobayashi, Y. Sasaki, H. Kobayashi, G. Saito, and H. Inokuchi, Bull. Chem. Soc. Jpn. 57, 627 (1984). 10.1246/bcsj.57.627 Crossref, Google Scholar
• 12 P. E. Blöchl, Phys. Rev. B 50, 17953 (1994). 10.1103/PhysRevB.50.17953 Crossref, Google Scholar
• 13 A. Dal Corso, Comput. Mater. Sci. 95, 337 (2014). 10.1016/j.commatsci.2014.07.043 Crossref, Google Scholar
• 14 P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. Küçükbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V. Nguyen, A. O. de-la Roza, L. Paulatto, S. Poncé, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu, and S. Baroni, J. Phys.: Condens. Matter 29, 465901 (2017). 10.1088/1361-648X/aa8f79 Crossref, Google Scholar
• 15 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 10.1103/PhysRevLett.77.3865 Crossref, Google Scholar
•   (16) A voltage terminal was also used as a current probe because the corresponding current lead was disconnected during pressure application. Google Scholar
•   (17) The crystallographic data at 296 K are: formula C14H8S8CuN6, space group C2/c, Z = 4, a = 16.756(3) Å, b = 13.975(3) Å, c = 10.340(2) Å, β = 124.145(10)°, V = 2003.9(7) Å3, dcalc = 1.923 g cm−3. R1[for I > 2σ(I), 1882 reflections] = 0.041, wR2(for all 2306 reflections) = 0.1127, and Goodness-of-fit (GOF) = 1.099. The data at 100 K are: a = 16.781(3) Å, b = 13.816(3) Å, c = 10.3114(16) Å, β = 123.816(5)°, V = 1986.3(6) Å3, dcalc = 1.940 g cm−3. R1[for I > 2σ(I), 2094 reflections] = 0.0374, wR2(for all 2196 reflections) = 0.1184, and GOF = 1.332. Google Scholar
•   (18) In the 1st BZ, αk ≥ 0; thus, if βk ≥ 0, then: condition E±(k) = 0 is fulfilled by following two cases: (i) αk = βk = 0 or (ii) tαk + t′βk = 0 and cos(a′ka′ − b′kb′) − 1 = 0. We then obtain two nodal lines, as described in the text. Node (i) linearly connects Z with, e.g., \((\mathop {a'}\nolimits^{*}/2, - \mathop {b'}\nolimits^{*}/2,\mathop {c'}\nolimits^{*}/2)\), while node (ii) is bending through, e.g., \(( - \mathop {a'}\nolimits^{*}/2, - \mathop {b'}\nolimits^{*}/2,\mathop {c'}\nolimits^{*}/2)\). When βk < 0, it is numerically confirmed that there is no solution for E±(k) = 0. Google Scholar
• 19 A. Otsuka, Y. Shimizu, G. Saito, M. Maesato, A. Kiswandhi, T. Hiramatsu, Y. Yoshida, H. Yamochi, M. Tsuchiizu, Y. Nakamura, H. Kishida, and H. Ito, Bull. Chem. Soc. Jpn. 93, 260 (2020). 10.1246/bcsj.20190279 Crossref, Google Scholar
• 20 H.-L. Liu, L.-K. Chou, K. A. Abboud, B. H. Ward, G. E. Fanucci, G. E. Granroth, E. Canadell, M. W. Meisel, D. R. Talham, and D. B. Tanner, Chem. Mater. 9, 1865 (1997). 10.1021/cm970143t Crossref, Google Scholar
• 21 J. C. Bonner and M. E. Fisher, Phys. Rev. 135, A640 (1964). 10.1103/PhysRev.135.A640 Crossref, Google Scholar
• 22 W. E. Estes, D. P. Gavel, W. E. Hatfield, and D. J. Hodgson, Inorg. Chem. 17, 1415 (1978). 10.1021/ic50184a005 Crossref, Google Scholar