Magnetic field driven transition between valence bond solid and antiferromagnetic order in a distorted triangular lattice

A molecular Mott insulator $\kappa \text{−}{\left(\mathrm{ET}\right)}_2\mathrm{B}{\left(\mathrm{CN}\right)}_4$ [ET = bis(ethylenedithio)tetrathiafulvalene] with a distorted triangular lattice exhibits a quantum disordered state with gapped spin excitation in the ground state. ${}^{13}\mathrm{C}$ nuclear magnetic resonance, magnetization, and magnetic torque measurements reveal that magnetic field suppresses valence bond order and induces long-range magnetic order above a critical field $\sim 8$ T. The nuclear spin-lattice relaxation rate $1/T_1$ shows persistent evolution of antiferromagnetic correlation above the transition temperature, highlighting a quantum spin liquid state with fractional excitations. The field-induced transition as observed in the spin-Peierls phase suggests that the valence bond order transition is driven through renormalized one-dimensionality and spin-lattice coupling.

Figure 1

(a) Anisotropic triangular lattice of molecular dimers in $\kappa \text{−}{\left(\mathrm{ET}\right)}_2\mathrm{B}{\left(\mathrm{CN}\right)}_4$ with two kinds of transfers $t$ and $t^{\prime }$ above $T_c$. Spin-1/2 on each ET dimer is paired to form a valence bond crystal below $T_c=5\mathrm{K}$. (b) Magnetic phase diagram based on NMR (closed symbols) and spin susceptibility $\chi$ (open symbols). Circles and diamonds with the uncertainty denote the valence bond order and antiferromagnetic transition temperatures, respectively, which are obtained from the temperature dependence of nuclear spin-lattice relaxation rate $T_1^{-1}$.

Figure 2

(a) Spin susceptibility $\chi$ for a polycrystalline sample of $\kappa \text{−}{\left(\mathrm{ET}\right)}_2\mathrm{B}{\left(\mathrm{CN}\right)}_4$ at 1 T [33] after subtracting the core diamagnetic susceptibility (${\chi }_{\mathrm{dia}}=-4.52\times {10}^{-4}$ emu ${\mathrm{mol}}^{-1}$). The red curve is a fitting result by a high-temperature series expansion [37]. Inset: Temperature dependence of $\chi$ at low temperatures under the magnetic field of 0.5, 1, 3, and 5 T. (b) Inverse temperature dependence of $\chi$ for the evaluation of spin gap with the exponential function $\chi \sim \exp (-\mathrm{\Delta }/T)$.

Figure 3

(a) ${}^{13}\mathrm{C}$ NMR spectra at $H=3.9\mathrm{T}$ along the $a$ axis in $\kappa \text{−}{\left(\mathrm{ET}\right)}_2\mathrm{B}{\left(\mathrm{CN}\right)}_4$. Lorentzian fitting curves are shown at the lowest temperature. (b) Knight shifts, $K\left(1\right)$ and $K\left(2\right)$, for two carbon sites, C(1) and C(2), which are measured from the reference (tetramethylsilane). A green curve denotes the spin susceptibility $\chi$ at 1.0 T (the right-hand axis) [33].

Figure 4

${}^{13}\mathrm{C}$ Knight shifts plotted against $\chi$ for 10–50 K. The linearity yields the hyperfine coupling constant for two ${}^{13}\mathrm{C}$ sites. The $y$-intercept gives the chemical shift.

Figure 5

Magnetic torque $\tau$ divided by magnetic field $H$ at low temperatures. The data at each temperature include a constant offset for clarity.

Figure 6

(a) Magnetic field dependence of the ${}^{13}\mathrm{C}$ NMR spectrum for $\kappa \text{−}{\left(\mathrm{ET}\right)}_2\mathrm{B}{\left(\mathrm{CN}\right)}_4$ at 1.4 K. Frequency $\omega$ was converted to the local field $\mathrm{\Delta }B$ using a relation $\mathrm{\Delta }B=\mathrm{\Delta }\omega /{\gamma }_0$ (${\gamma }_0=10.7054$ MHz/T). (b) Definition of the 1/2 and 1/10 linewidths for $T=1.4\mathrm{K}$ and $H=12\mathrm{T}$, where the spectrum is constructed as a sum of Fourier transformed spin-each intensities taken with a 0.2 MHz step. (c) Magnetic field dependence of the 1/2 and 1/10 linewidths.

Figure 7

Temperature dependence of the ${}^{13}\mathrm{C}$ NMR spectrum for the single crystal of $\kappa \text{−}{\left(\mathrm{ET}\right)}_2\mathrm{B}{\left(\mathrm{CN}\right)}_4$ under a magnetic field of (a) 8.5 and (b) 14 T along the $a$ axis.

Figure 8

(a) Temperature dependence of the ${}^{13}\mathrm{C}$ nuclear spin-lattice relaxation rate $T_1^{-1}$ at 3.9–7.8 T. ${}^1\mathrm{H}$ NMR $T_1^{-1}$ (a long component) at 2.0 T scales to ${}^{13}\mathrm{C}$ data by multiplying ${10}^2$ [33]. $T_c$ defined by the onset of $T_1^{-1}$ drop is marked by arrows. (b) Inverse temperature dependence of $T_1^{-1}$ for evaluating the spin gap $\mathrm{\Delta }$. (c) Field dependence of $\mathrm{\Delta }$ (the right-hand axis) and the contour plot of $T_1^{-1}$ obtained from the temperature and field dependence measurement. Solid and open circles are evaluated from $T_1^{-1}$ and $\chi$, respectively.

Figure 9

(a) Temperature dependence of $T_1^{-1}$ above 7.8 T. Arrows point to the magnetic order transition at $T_{\mathrm{N}}$. (b) Linewidth of the ${}^{13}\mathrm{C}$ NMR spectrum, defined by 1/10 of the peak intensity in Fig. 6.

Figure 10

Temperature dependence of $T_1^{-1}$ for ${}^{13}\mathrm{C}$ NMR in $\kappa \text{−}{\left(\mathrm{ET}\right)}_{2}X$, $X=\mathrm{B}{\left(\mathrm{CN}\right)}_4$, ${\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ [42], and ${\mathrm{Ag}}_2{\left(\mathrm{CN}\right)}_3$ [32].