A molecular diamond lattice antiferromagnet as a Dirac semimetal candidate

TL;DR

This study investigates a molecular diamond-lattice compound, revealing antiferromagnetic order at high temperature and potential as a Dirac semimetal, with implications for topological materials.

Contribution

It demonstrates that (ET)Ag$_4$(CN)$_5$ exhibits high-temperature antiferromagnetic order and is a candidate for a three-dimensional topological Dirac semimetal.

Findings

Antiferromagnetic order at 102 K with weak ferromagnetism.

Transition temperature increases to ~195 K under 2 GPa pressure.

First-principles calculations suggest Dirac semimetal potential.

Abstract

The ground state of a molecular diamond-lattice compound (ET)Ag$_4$(CN)$_5$ is investigated by the magnetization and nuclear magnetic resonance spectroscopy. We found that the system exhibits antiferromagnetic long-range ordering with weak ferromagnetism at a high temperature of 102 K owing to the strong electron correlation. The spin susceptibility is well fitted into the diamond-lattice Heisenberg model with a nearest neighbor exchange coupling of 230 K, indicating the less frustrated interactions. The transition temperature elevates up to $\sim$195 K by applying pressure of 2 GPa, which records the highest temperature among organic molecular magnets. The first-principles band calculation suggests that the system is accessible to a three-dimensional topological semimetal with nodal Dirac lines, which has been extensively searched for a half-filling diamond lattice.

Full text

A half-filling diamond lattice has been recently attracted great interests as an example of three-dimensional (3D) Dirac semimetal with the linearly-crossing band dispersion near the Fermi level Fu et al. (2007); Zhang et al. (2009); Young et al. (2012); Ryu (2009) along with the appealing example including Na3Bi and Cd3As2 Liu et al. (2014); Borisenko et al. (2014). The system can be a strong topological insulator in the presence of spin-orbit coupling as a 3D analogue to graphene Fu et al. (2007). Despite the popular crystal structure, the material with the half-filled band has been known only in a putative material BiO2 Young et al. (2012). In the counterpart insulating system, the frustrated local moment on the diamond lattice has been extensively studied as a spin liquid candidate Bergman et al. (2007); Bernier et al. (2008); Buessen et al. (2018). A typical example is the magnetic spinel ($AB_{2}C_{4}$) with the $A$ site diamond lattice Fritsch et al. (2004); Krimmel et al. (2009); Plumb et al. (2016); Ge et al. (2017); Chamorro et al. (2018); Iakovleva et al. (2015); MacDougall et al. (2016, 2011); Zaharko et al. (2011), where the properties of the disordered state are under intense debate. A (topological) Mott transition is expected to occur from a spin disordered phase to a Dirac semimetal phase by tuning the electron correlation Zhang et al. (2009); Bergman et al. (2007).

Organic molecular compounds have provided the platform for investigating the pressure-tuned Mott transition for the soft crystal. The well-studied Mott-Hubbard systems such as $\kappa$-(ET)2X and Z[Pd(dmit)2]2 possess a quasi-two-dimensional triangular lattice of the molecular dimer unit Shimizu et al. (2003); Itou et al. (2010); Isono et al. (2014) owing to the anisotropic intermolecular interactions between planar molecules. Thus there are only a few example of 3D molecular compounds including the diamond lattice, except for the inorganic-organic hybrid system such as Li(TCNE) and Cu(DCNQI)2 Kareis et al. (2012); Her et al. (2013); Ermer (1991), and no example is known for the half-filling diamond lattice consisting of organic molecules. The material search for 3D molecular compounds would be important for giving high-temperature magnets and superconductors.

We present here the molecular material (ET)Ag4(CN)5 Geiser et al. (1985) as a prime example of the 3D diamond lattice. It possesses the extremely high-symmetry crystal structure of the orthorhombic $Fddd$ lattice with the lattice constants: $a=13.215(9)$ Å, $b=19.4783(1)$ Å, and $c=19.6506(1)$ Å. Each monovalent ET molecule is surrounded by the honeycomb framework of the closed-shell polyanion [Ag4(CN)5]${}_{\infty}^{-}$ in the $bc$ plane, as shown in Fig. 1(a). The next stacking layer along the $a$ axis is deviated by [$1/4$, $1/4$, $1/4$] in the unit cell. This unique packing pattern is distinct from the layered or columnar structure in typical molecular solids. The alternating stacking makes ET molecules to construct a diamond lattice with four equivalent nearest neighbor transfers. Therefore, the system is regarded as a half-filling diamond lattice of the molecular unit [Fig. 1(b)], which can be either a spin-$1/2$ Heisenberg antiferromagnet or a 3D Dirac semimetal, depending on the strength of electron correlations.

In this Letter, we investigate the ground state of the diamond-lattice molecular compound (ET)Ag4(CN)5 through the resistivity, magnetization, and nuclear magnetic resonance (NMR) measurements. As expected for the half-filling organics composed of the monovalent ions, the ground state behaves as a Mott insulator with a spin-$1/2$ on each molecule. We determined the spin structure and dynamics through the angular dependence of 13C NMR at ambient pressure. Together with the band calculation, we discuss the possible Dirac semimetal phase emerging from the antiferromagnetically ordered phase.

Single crystals of (ET)Ag4(CN)5 were prepared by galvanostatic electrooxidation of ET in a 1,1,2-trichloroethane solution of KAg(CN)2 and 18-crown-6 ether. The obtained rhombohedral shaped crystals of (ET)Ag4(CN)5 were carefully separated from minor co-products of $\kappa$-(ET)2Ag2(CN)3 Shimizu et al. (2016), $\alpha^{\prime}$-(ET)2Ag(CN)2 Beno et al. (1986), and $\kappa$-(ET)2Ag(CN)2 $\cdot$ H2O Kurmoo et al. (1988). Resistivity was measured with a four-probe dc method at ambient and hydrostatic pressures, where gold wires were attached to a single crystal using the carbon paint. Hydrostatic pressure was applied using a BeCu piston cylinder cell with Daphne 7373 oil and monitored by the manganin wire resistance. Magnetic susceptibility was measured for a polycrystalline sample by a superconducting quantum interference device magnetometer (Quantum Design MPMS-XL). The core diamagnetism value was calculated as a sum of Pascal’s constants ($-3.74\times 10^{-4}$ emu mol*-1*). 1H and 13C NMR spectra were obtained for a single crystal in a static magnetic field of 2.0 T and 8.5 T, respectively, which were calibrated using the resonance frequency of the standard sample, tetramethylsilane (TMS).

The band structure of (ET)Ag4(CN)5 was obtained from the first-principles density-functional-theory (DFT) calculation based on the generalized gradient approximation with the WIEN2K code, as shown in Fig. 1(c). Here the band structure was evaluated without considering the anion having the orientational CN/NC disorder (Fig. S2) SI . The Fermi energy is located at the half-filling position, where two HOMO bands cross. The two bands are degenerate along $Z-X$ and $Y-X$ directions, characteristic in the diamond lattice Chadi and Cohen (1975). The result qualitatively agrees with the extended H${\rm\ddot{u}}$ckel and the tight-binding calculation Geiser et al. (1985, 1987). Along the symmetry positions of the band, nodal Dirac lines appear near the Fermi level, as shown in Fig. 1(d), consistent with the result for the cubic diamond lattice Young et al. (2012).

Despite the semimetallic band structure, the resistivity $\rho$ exhibits insulating temperature ($T$) dependence, as shown in Fig. 2(a). The activation energy is obtained as $E_{b}$ = 0.15 eV along the $b$ axis at ambient pressure. Together with the paramagnetic spin susceptibility $\chi$ [Fig. 2(b)], the system is regarded as a Mott insulator due to the on-site Coulomb interaction $U(\sim 1$ eV for the monovalent ET) greater than the bandwidth ($W\sim 0.57$ eV) [Fig. 1(c)]. By applying hydrostatic pressure, $\rho$ is suppressed by an order of magnitude, and $E_{b}$ decreases to 0.12 eV at 1.5 GPa.

The temperature dependence of $\chi$ [Fig. 2(b)] is distinct from the Curie-Weiss law in classical paramagnets, but exhibits a broad maximum around 220 K. The weak temperature dependence of $\chi$ is similar to the triangular-lattice anitiferromagnet such as $\kappa$-(ET)2Cu2(CN)3 Shimizu et al. (2003), and the 1D Heisenberg antiferromagnet such as Sr2CuO3 Motoyama et al. (1996), suggesting significant quantum fluctuations. The experimental data are well fitted by the high-temperature series expansion of the $S=1/2$ diamond-lattice Heisenberg model with the Pad${\rm\acute{e}}$ approximant [7, 7] Oitmaa (2018), yielding the antiferromagnetic exchange coupling $J=230\pm 10$ K. An indication of the magnetic transition is observed at $T_{\rm N}$ = 102 K, where $\chi$ shows an abrupt increase at low fields. The prominent magnetic field ($H$) dependence in the magnetization $M$ and the thermal hysteresis [Fig. 2(b) inset, Fig. S3] highlight weak-ferromagnetism due to the canting of antiferromagnetic moments.

Microscopic evidence for long-range magnetic ordering is given by 1H and 13C NMR measurements on the single crystal of (ET)Ag4(CN)5, as shown in Fig. 3. Above 105 K, the 1H NMR spectrum [Fig. 3(a)] represents the 1H-1H nuclear dipole coupling with the $T$-independent linewidth ($\sim 50$ kHz). The spectrum begins to split into four below 102 K. The splitting develops upon cooling temperature, signaling the emergence of huge local fields parallel and antiparallel to the external field. The presence of two inequivalent 1H sites on an ET molecule further splits each spectrum into two.

To determine the ordered spin texture, the 13C NMR spectrum was measured for the isotope enriched crystal at the central double-bonded C sites with the high electron density Larsen and Lenoir (1988). There is only a single 13C site manifested as a sharp line in the paramagnetic state. The linewidth grows as spin fluctuations slow down toward $T_{\rm N}$. Below $T_{\rm N}$, the spectrum splits into two due to the antiferromagnetic order, consistent with 1H NMR. The angular dependence of the NMR frequency $\omega$ around the crystal axes [Fig. 3(c,d)] shows that the local field ${\bf H}_{\rm loc}$ exhibits a minimum or maximum against the external magnetic field ${\bf H}_{0}$ parallel to ${\bf b\pm a}$. A rotation of ${\bf H}_{0}$ from the ${\bf b-a}$ to ${\bf c}$ confirms ${\bf H}_{\rm loc}\parallel{\bf b\pm a}$ [Fig. 3(d)].

Here the 13C nuclear spin experiences a sum of the external field ${\bf H}_{0}$ and the spontaneous local field ${\bf H}_{\rm loc}$ produced by the magnetic moment ${\bf M}_{\rm loc}$: $\omega$ is given by $\omega=\gamma_{n}H_{\rm eff}=\gamma_{n}(|{\bf H}_{0}+{\bf H}_{\rm loc}|)=\gamma_{n}\sqrt{H_{0}^{2}+H_{n}^{2}+2H_{0}H_{n}{\rm cos}\vartheta}$, where $\gamma_{n}$ is the 13C nuclear gyromagnetic ratio ($10.7054$ MHz/T) and $\vartheta$ is the angle between ${\bf H}_{0}$ and ${\bf H}_{\rm loc}$. ${\bf H}_{\rm loc}$ is given by ${\sf A}{\bf M}_{\rm loc}=(A_{aa}M_{a},A_{bb}M_{b},A_{cc}M_{c})$ using the hyperfine coupling tensor ${\sf A}$ with diagonal components $A_{\alpha\alpha}(\alpha=a,b,c)$ and ${\bf M}_{\rm loc}=(M_{a},M_{b},M_{c})$. We determined ${\sf A}$ from the $K-\chi$ plot (Fig. S5) as $(A_{aa},A_{bb},A_{cc})=(-0.21.-0.21,1.64)$ T/$\mu_{\rm B}$, which yields the isotropic Fermi contact ($\alpha=0.41$ T/$\mu_{\rm B}$) and the anisotropic dipole hyperfine coupling ($\beta=0.62$ T/$\mu_{\rm B}$) as expected for the $sp_{2}$ orbital Miyagawa et al. (2004).

The temperature dependence of 13C NMR spectrum for ${\bf H}_{0}\parallel[110]$ [Fig. 3(b)] demonstrates the evolution of the order parameter, following a scaling law $\sim(T_{\rm N}-T)^{\beta}$ with a critical exponent $\beta=0.36\pm 0.02$ for $T>70$ K [Fig. 3(b)]. It is in good agreement with $\beta$ = 0.368 in the 3D Heisenberg model Benner and Boucher (1990). By using the obtained ${\sf A}$, we determined the magnetic moment ${\bf M}_{\rm loc}=(0.114,0.889,0.0)\mu_{\rm B}$ with the magnitude $|{\bf M}_{\rm loc}|=0.90\pm 0.02\mu_{\rm B}$ ($\mu_{\rm B}$: the Bohr magneton) at 40 K. Namely, the easy axis of the moment is directed close to the $b$ axis with the collinear configuration, as schematically shown in Fig. 4(d), consistent with the theoretical ground state for the less frustrated diamond lattice Bergman et al. (2007); Bernier et al. (2008). The tiny canting of the moment ($\sim 0.012^{\circ}$ obtained from the $M-H$ curve, Fig. S3) was not detected within the accuracy of the NMR measurement.

Interamolecular interactions can be sensitively tuned for the soft organic crystal by applying pressure. Figure 4 shows the 13C Knight shift $K$ and the nuclear spin-lattice relaxation rate $T_{1}^{-1}$ for (ET)Ag4(CN)5 under hydrostatic pressure up to 2 GPa. Here $K$ is converted into the local spin susceptibility $\chi$ using a relation $K=A\chi/(N\mu_{\rm B})$, $N$: the Avogadro number) with the hyperfine coupling constant $A$ at ambient pressure [Fig. 4(a)], whereas $T_{1}^{-1}$ measures the dynamical spin susceptibility that scales to $J^{-1}$ at high temperatures: $T_{1}^{-1}=\sqrt{\frac{\pi}{3}}\frac{A^{2}\sqrt{S(S+1)}}{\hbar J\sqrt{z}}$ Moriya (1956). We obtained $J$ = 240$\pm 20$ K at 0 GPa, in agreement with that obtained from $\chi$. By applying pressure, $\chi$ is suppressed owing to an increase in $J$. Fitting of $\chi$ into the diamond-lattice Heisenberg model Oitmaa (2018) allows a rough estimate of the exchange coupling: $J=310\pm 40$ K at 1.0 GPa and $400\pm 30$ K at 2.0 GPa.

Simultaneously, $T_{\rm N}$ obtained from the spectral splitting [Fig. 4(b)] and the sharp $T_{1}^{-1}$ peak [Fig. 4(c)] elevates as we increase the pressure: $T_{\rm N}=150$ K at 1.0 GPa and 195 K at 2.0 GPa [Fig. 4(d)]. It corresponds to the highest magnetic transition temperature among molecular materials including the organic charge-transfer salt Smith et al. (2004); Miyagawa et al. (2004), the transition-metal hybrid system such as [Au(tmdt)2] ($T_{\rm N}\sim 110$ K) Hara et al. (2008), and the C60 complex such as (NH3)KRb2C60 ($T_{\rm N}=76$ K) Takenobu et al. (2000). In contrast to the mean-field theory giving $T_{\rm N}\sim J=\Theta$ ($\Theta$: Weiss temperature) for the diamond lattice, the experimentally obtained $T_{\rm N}$ is suppressed to the temperature scale of $\sim J/2$. It is consistent with the significant quantum fluctuations in the $S=1/2$ Heisenberg antiferromagnet Oitmaa (2018). Despite an increase of $T_{\rm N}$, the magnetic moment is suppressed upon increasing pressure [Fig. 4(b)]. The moment contraction is attributable to quantum fluctuations due to the electron itinerancy and a weak dimerization of ET molecules.

Theoretically, the diamond lattice involves geometrical frustration due to 12 next nearest neighbor interactions $J^{\prime}$. An introduction of the small $J^{\prime}\sim J/8$ can suppress $T_{\rm N}$ and induce a spin liquid state Bergman et al. (2007); Bernier et al. (2008). The frustration is released by strong thermal or quantum fluctuations via order-by-disorder mechanism, where the ground state is determined by an entropical or energetical selection Bergman et al. (2007); Bernier et al. (2008). In the present case, however, a tight-binding calculation suggests the negligible $J^{\prime}/J\sim(t^{\prime}/t)^{2}<0.004$. Indeed, the obtained $T_{\rm N}/J=0.46$ at 0 GPa and 0.49 at 2.0 GPa are consistent with the $S=1/2$ diamond-lattice Heisenberg model including only the nearest neighbor interaction Oitmaa (2018). Furthermore, in the real system, the highly degenerated (six-fold) spin structure on the diamond lattice should be lifted by single-ion anisotropy (spin-orbit coupling) or lattice distortion (spin-phonon coupling), triggering the magnetic order. Whereas the $g$-value is nearly isotropic ($g_{a}=2.0026$, $g_{b}=2.0157$, $g_{c}=2.0069$) in (ET)Ag4(CN)5, the spin-orbit coupling as well as the spin-phonon coupling, which leads to Dzyaloshinskii-Moriya interaction, may play a key role in the spin texture. Despite the presence of the structural CN/NC disorder in the anion, analogous to $\kappa$-(ET)${}_{2}X_{2}$(CN)3 ($X$ = Ag, Cu) without magnetic ordering Shimizu et al. (2003, 2016), the present system with a less frustrated diamond lattice exhibits the high $T_{\rm N}$. It suggests that the disorder potential from the counter ion plays an negligible effect in the magnetic ground state.

Our finding demonstrates the potential of the molecular conductors for three-dimensional and high-transition temperature magnets through the combination of polymeric counter ions. The high-$T_{\rm N}$ Mott insulator may host high-$T_{\rm c}$ superconductivity across the Mott transition under high pressure, because the energy scale of the exchange interaction ($>$ 400 K) for (ET)Ag4(CN)5 may be greater than those of the dimer ET salts such as $\beta^{\prime}$-(ET)2ICl2 ($T_{c}$ = 14.2 K at 8.2 GPa) Taniguchi et al. (2003) and C60 complex such as Cs3C60 ($T_{\rm c}$ = 38 K) Takabayashi et al. (2009); Takenobu et al. (2000). As shown by the band calculation, the metallic phase induced by high pressure may be 3D Dirac semimetal, which has a topological surface state in the presence of spin-orbit coupling and inversion symmetry breaking Fu et al. (2007); Young et al. (2012). The emergence of the Dirac semimetal has recently observed in a single-component molecular conductor under high pressures Kato et al. (2017); Liu et al. (2018).

To conclude, we investigated the ground state for the uniquely high-symmetry organic Mott insulator with the diamond lattice, (ET)Ag4(CN)5, which possesses nodal Dirac lines in the original band structure without electron correlations. Whereas the charge activation energy exceeds 0.1 eV at ambient pressure, the antiferromagnetic exchange interaction reaches $J$ = 230 K, and the long-range magnetic order with the weak ferromagnetism occurs at the high temperature of 102 K. Furthermore, the application of hydrostatic pressure enhances the transition temperature up to 195 K, which is highest among the molecular systems and thus anticipated to host the high-$T_{c}$ superconductivity at higher pressure.

This work was supported by JSPS KAKENHI Grant Numbers JP16K13836, JP17H05151, and JP16H04012, JP16H04139, and JP26288035.