Pulsed-field magnetization, electron spin resonance, and nuclear spin-lattice relaxation in the $\left\{{\mathrm{Cu}}_3\right\}$ spin triangle

We report on pulsed-field magnetization, $Q$-band electron spin resonance (ESR), and ${}^{23}\mathrm{Na}$ NMR measurements of the $S=1∕2$ spin triangle clusters ${\mathrm{Na}}_9\left[{\mathrm{Cu}}_3{\mathrm{Na}}_3{\left({\mathrm{H}}_2\mathrm{O}\right)}_9{\left(\alpha \text{−}X{\mathrm{W}}_9{\mathrm{O}}_{33}\right)}_2\right]$ ($X=\mathrm{As}$ and Sb). The pulsed-field magnetization shows pronounced hysteresis loops and magnetization steps including the half-step magnetization. The detailed magnetization behavior depends substantially on the diamagnetic heteroatom $X$. The angular dependence of ESR parameters necessitates Dzyaloshinskii-Moriya interaction. The temperature dependence of the ${}^{23}\mathrm{Na}$ spin-lattice relaxation rate, $1∕T_1$, scales well to $\chi \left(T\right)T$, where $\chi \left(T\right)$ is the static susceptibility. The spin-spin relaxation rate, $1∕T_2$, increases rapidly for temperatures below $15\mathrm{K}$ due to dipolar interactions between the ${}^{23}\mathrm{Na}$ nuclei and ${\mathrm{Cu}}^{2+}$ spins. The two clusters exhibit a markedly different field dependence of $1∕T_1$ at antilevel crossing points. The enhancement of $1∕T_1$ is noticeable only for $X=\mathrm{Sb}$. Since the spin configurations of both clusters are nearly the same, the dependence of magnetization and $1∕T_1$ on $X$ is ascribed to the strong coupling of the spins to a lattice vibration, leading to an enhanced mixing of the $S=1∕2$ chiral state.

Figure 1

(Color online) (a) Combined polyhedral/ball-and-stick representation of ${\mathrm{Na}}_9\left[{\mathrm{Cu}}_3{\mathrm{Na}}_3{\left({\mathrm{H}}_2\mathrm{O}\right)}_9{\left(\alpha \text{−}X{\mathrm{W}}_9{\mathrm{O}}_{33}\right)}_2\right]$. Yellow ball is for Na atoms, cyan for Cu, green for $X$, and red for ${\mathrm{H}}_2\mathrm{O}$. [(b) and (c)] Sketch of the $\left\{{\mathrm{Cu}}_3\text{−}X\right\}$ spin triangle configuration with $\mathrm{Cu}\text{⋯}\mathrm{Cu}$ distances. ${\mathrm{Cu}}_1$ and ${\mathrm{Cu}}_2$ represent two crystallographically inequivalent Cu sites. The numbers on the solid lines are $\mathrm{Cu}\text{⋯}\mathrm{Cu}$ distances.

Figure 2

(Color online) (a) Magnetization curve vs pulsed magnetic field for $\left\{{\mathrm{Cu}}_3\text{−}\mathrm{As}\right\}$ at $0.4\mathrm{K}$ for $H\parallel$ triangle plane. The saturated magnetization is normalized by $gS$. Arrows are a guide to sweep directions $(A\to B\to C\to D)$. (b) Magnetization of $\left\{{\mathrm{Cu}}_3\text{−}\mathrm{Sb}\right\}$ using a pulsed field with the same conditions as (a). The inset shows the time dependence of a pulsed magnetic field.

Figure 3

Angular dependence of ESR spectra (derivative of the absorption spectra versus field, vertically shifted for clarity) measured at $\nu =34\mathrm{GHz}$ and $8.8\mathrm{K}$ for $\left\{{\mathrm{Cu}}_3\text{−}\mathrm{As}\right\}$. The angle is measured between the $C_3$ axis of a triangle and the external field. The three strong peaks of the right panel correspond to the transitions between the $S^T=3∕2$ states. The weak peaks arise from the transitions between the $S^T=1∕2$ states. Note the different magnetic field scales of the left and right panels, and ESR spectra of the left panel are ten times magnified.

Figure 4

(a) Temperature dependence of ESR intensity of the $\mid 7⟩\to \mid 8⟩$ (full triangle) and $\mid 6⟩\to \mid 7⟩$ (open rectangle) transitions. The inset displays the respective $g$-factor as a function of temperature. [(b) and (c)] Angular dependence of the resonance fields. The open (full) symbols represent the experimental data of $\left\{{\mathrm{Cu}}_3\text{−}\mathrm{As}\right\}$ $\left(\left\{{\mathrm{Cu}}_3\text{−}\mathrm{Sb}\right\}\right)$.

Figure 5

${}^{23}\mathrm{Na}$ NMR spectrum obtained by monitoring the FFT sum of a spin echo as a function of an external field at a fixed temperature of $1.82\mathrm{K}$. The shaded area indicates the ${}^{23}\mathrm{Na}$ NMR spectrum, which consists of the $+1∕2\leftrightarrow -1∕2$ central transition (marked P) and the quadrupolar wings. The empty solid line is ${}^{63}\mathrm{Cu}$ NMR spectrum from the rf coil. The $T_1$ measurements were made on the peak P.

Figure 6

(a) ${}^{23}\mathrm{Na}$ spin-lattice relaxation rate $(1∕T_1)$ of $\left\{{\mathrm{Cu}}_3\text{−}\mathrm{As}\right\}$ at $2\mathrm{T}$ (full square) and $4.4\mathrm{T}$ (open triangle) as a function of temperature. The solid line is the temperature dependence of $\chi \left(T\right)T$ at $4.4\mathrm{T}$. (b) Temperature dependence of spin-spin relaxation rate, $1∕T_2$, (full triangle) and ${}^{23}\mathrm{Na}$ FWHM (open square) for $\left\{{\mathrm{Cu}}_3\text{−}\mathrm{As}\right\}$ at $4.4\mathrm{T}$.

Figure 7

Field dependence of spin-lattice relaxation rate $1∕T_1$ at $1.67\mathrm{K}$ for $\left\{{\mathrm{Cu}}_3\text{−}\mathrm{As}\right\}$ compound and at $1.72\mathrm{K}$ for $\left\{{\mathrm{Cu}}_3\text{−}\mathrm{Sb}\right\}$ compound, respectively. The solid line is a fit of the data to Eq. (2). Inset: Field dependence of $1∕T_1$ subtracted by the background, which is obtained by Eq. (3). The solid line is a calculated curve with $\Delta =0.6\mathrm{K}$.

Figure 8

(Color online) Energy level diagram for $\left\{{\mathrm{Cu}}_3\text{−}\mathrm{As}\right\}$. Field dependence of the eight eigenvalues is obtained by solving the Hamiltonian (4) numerically.

Figure 9

(Color online) A schematic of magnetization following energy levels 1 and 2 in an upward sweep. See the text for details.