Molecular diamond lattice antiferromagnet as a Dirac semimetal candidate

The ground state of a molecular diamond lattice compound $(\mathrm{ET}){\mathrm{Ag}}_4{\left(\mathrm{CN}\right)}_5$ is investigated by magnetization and nuclear magnetic resonance spectroscopy. We found that the material exhibits a Mott-insulating ground state with antiferromagnetic long-range ordering at 102 K. The ordered moment shows weak ferromagnetism with a tiny canting angle. 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 $\sim 195\mathrm{K}$ by applying pressure of 2 GPa, which records the highest temperature among organic molecular magnets. The first-principles band calculation without electron correlations suggests that the system is accessible to a three-dimensional topological semimetal with nodal Dirac lines, which has been anticipated on a half-filling diamond lattice.

Figure 1

(a) Crystal structure of ($\mathrm{ET}){\mathrm{Ag}}_4{\left(\mathrm{CN}\right)}_5$ with a diamond lattice. Each molecule connected by the symmetry operation [$1/4,\pm 1/4,\mp 1/4$] has four equivalent nearest-neighbor interactions (dotted lines). (b) Diamond lattice of ET where the red sphere represents the centroid of ET. (c) Highest-occupied molecular orbital (HOMO) bands based on the first-principles calculation. The degenerate band crosses the Fermi level along $Z-X-Y$ due to second neighbor transfers, which gives electron and hole-like Fermi surfaces [Fig. 6]. Dotted lines are the fitting results with tight-binding parameters of transfer integrals (see the Appendix). The density of states (DOS) has the V-shaped energy dependence centered at the Fermi energy. (d) Three-dimensional projection of the band dispersion, showing nodal Dirac lines along the symmetry points along $Z-X-Y$.

Figure 2

(a) Inverse temperature dependence of resistivity $\rho$ measured along the $b$ axis of $\left(\mathrm{ET}\right){\mathrm{Ag}}_4{\left(\mathrm{CN}\right)}_5$ at 0.0, 0.5, 1.0, and 1.5 GPa. (b) Temperature dependence of the magnetic susceptibility $\chi$ for zero-field cooling (ZFC) at $H=0.01$, 0.2, and 5.0 T, where the core diamagnetic contribution was subtracted. The solid curve is a fitting result using a diamond lattice Heisenberg model [31] with the $\mathrm{Pad}\acute{\mathrm{e}}$ approximant [7, 7] and $J=230\mathrm{K}$. Inset: ZFC and FC magnetization at 0.01 T.

Figure 3

(a) ${}^1\mathrm{H}$ NMR spectrum along the $a$ axis at 2.0 T. (b) ${}^{13}\mathrm{C}$ NMR spectrum at 8.5 T applied along [110] corresponding to the magic angle where the ${}^{13}\mathrm{C}-{}^{13}\mathrm{C}$ dipole coupling vanishes for $T>T_{\mathrm{N}}$. Dotted curves represent temperature dependence of order parameters with the critical exponent $\beta =0.36$, as expected in 3D Heisenberg antiferromagnets. The vertical axis scales to temperature. (c, d) Angle dependence of ${}^{13}\mathrm{C}$ NMR frequency in the ordered state at 95 K, where $\phi$ and $\theta$ are defined by the angles measured from the $-a$ and $-a+b$ directions toward the $b$ and $c$ axes, respectively.

Figure 4

(a) Static spin susceptibility $\chi$ obtained from ${}^{13}\mathrm{C}$ Knight shift $K=A\chi /\left(N{\mu }_{\mathrm{B}}\right)$ with the hyperfine coupling constant $A=0.83\mathrm{T}/{\mu }_{\mathrm{B}}$ determined at 0 GPa. The magnetic field was applied parallel to [110]. Solid curves are the series expansion of the diamond lattice Heisenberg model [31]. (b) ${}^{13}\mathrm{C}$ NMR frequency defined by the spectral peak across $T_{\mathrm{N}}$ at 0, 1.0, and 2.0 GPa, where the magnetic field is applied along [110]. (c) The nuclear spin-lattice relaxation rate $T_1^{-1}$. The steep increase toward magnetic ordering occurs due to slowing down to spin fluctuations. (d) Pressure-temperature phase diagram of ($\mathrm{ET}){\mathrm{Ag}}_4{\left(\mathrm{CN}\right)}_5$ based on the NMR measurements. The inset shows the spin configuration determined by ${}^{13}\mathrm{C}$ NMR.

Figure 5

Diffuse x-ray scattering images for a single crystal of ($\mathrm{ET}){\mathrm{Ag}}_4{\left(\mathrm{CN}\right)}_5$. The data were taken at 300, 110, 40, and 8.5 K for the $ac$ plane ($a^{*}{c}^{*}$ for the reciprocal lattice). A superstructure develops along the $c$ axis upon cooling.

Figure 6

First-principles band structure calculations. (a) A full band structure of ($\mathrm{ET}){\mathrm{Ag}}_4{\left(\mathrm{CN}\right)}_5$. The Fermi energy ($E=0$) is located at the half-filled position of the HOMO band. (b) Electron (blue) and hole (red) Fermi surfaces in the 3D Brillouin zone of the diamond lattice [44].

Figure 7

Magnetization-magnetic field ($M-H$) curves in ($\mathrm{ET}){\mathrm{Ag}}_4{\left(\mathrm{CN}\right)}_5$. (a) A low-field cycle between $\pm 500\mathrm{Oe}$ at different temperatures. An anomaly at 150 Oe and 1.9 K is an extrinsic origin due to the measurement across zero magnetization in raw signals before subtracting the diamagnetic contribution. (b) Magnetic field cycle from $-50$ to 50 kOe at 1.9 K. A curve starts from zero field and then turns at 50 and $-50\mathrm{kOe}$.

Figure 8

Angular dependence of ${}^{13}\mathrm{C}$ NMR spectrum. The data were taken for a single crystal of ($\mathrm{ET}){\mathrm{Ag}}_4{\left(\mathrm{CN}\right)}_5$ at 8.5 T and 290 K. (b) Temperature dependence of the ${}^{13}\mathrm{C}$ Knight shift $K$ in ($\mathrm{ET}){\mathrm{Ag}}_4{\left(\mathrm{CN}\right)}_5$ for $H\parallel a,b$ and $c$ at $H=8.50\mathrm{T}$. (b) $K$ versus $\chi$ plots for the crystal axes, where the linearity coefficient gives the hyperfine coupling constant.