Slow Spin Dynamics in the Hyper-Honeycomb Lattice [(C₂H₅)₃NH]₂Cu₂(C₂O₄)₃ revealed by ¹H NMR Studies

Magnetic frustration and quantum fluctuations, maximized for low spin $S$ = 1/2, are sources of a variety of fascinating phenomena LB ; PA ; SS ; CL ; HTD . One of the exotic physical phenomena is a quantum spin-liquid state which breaks no symmetries down to zero temperature but exhibits macroscopic entanglement of strongly interacting spins and features exotic fractionalized excitations LB ; PA . Following theoretical progress in understanding the quantum state, more and more candidate spin-liquid materials have been discovered and extensively studied owing to advancements in material synthesis and experimental characterization techniques. The most prominent spin-liquid candidates reported so far are $S$ = 1/2 kagomé lattices ZnCu₃(OH)₆Cl₂ TH2012 ; PM1_2011 ; JS2007 ; AO ; TI ; BF ; PMRev ; ImaiSci , [NH₄]₂[C₇H₁₄N][V₇O₆F₁₈] LC , S = 1 hyperkagome Na₃Ir₄O₈ YO , and $S$ = 1/2 triangular lattice organic compounds EtMe₃Sb[Pd(dmit)₂]₂ RK and $\mathit{\kappa}$-(BEDT-TTF)₂Cu₂(CN)₃ YSZ ; FLP ; SY ; SYQ .

Recently, a possible spin-liquid state has been reported in [(C₂H₅)₃NH]₂Cu₂(C₂O₄)₃ (hereafter, CCCO) CuOx where Cu²⁺ ($S$ =1/2) ions form a three-dimensional hyper-honeycomb lattice or a ths net in the Reticular Chemistry Structure Resource code notation RCSR . The schematic arrangement of Cu²⁺ ($S$ = 1/2) is shown in the inset of Fig. 1 where the two different Cu sites (Cu1 and Cu2) form zigzag chains along the $c$ axis and the $a$ axis direction, respectively, and produce the four different magnetic interactions of $J_{1}$, $J_{2}$, $J_{3}$ and $J_{4}$ (for more detail, see supplementary material, SM SI ). Although the magnetic interaction between Cu²⁺ spins in the system seems to be complicated due to the structure and exchange paths, the maximum antiferromagnetic (AFM) interaction of $J_{4}$ is reported to be $\sim$ 200 K with $J_{4}~{}>~{}J_{1}\sim 60~{}$K$~{}>~{}J_{2}\sim 25~{}$K$~{}>~{}J_{3}$ in Ref. CuOx . Nevertheless, no sign of long-range magnetic-order down to a temperature $T$ = 60 mK has been observed by the $\mu$SR measurements CuOx . The specific heat ($C$) measurement shows a relatively large value of $\gamma$ = 36 mJ/(Cu-mol$\cdot$K²) for the linear term of $C$/$T$, indicating the gapless spin excitation in the ground state. Based on those experimental results, Zhang ${\it et~{}al.}$ CuOx suggested the ground state of the compound is a gapless spin-liquid state. On the other hand, the quite recent density-functional theory (DFT) calculations jacko21 suggested that the ground state of the compound may not be a spin-liquid state but can be regarded as a weakly-coupled AFM Heisenberg chain system. Despite the current interest in the compound as a spin-liquid candidate, the ground state of CCCO is still an open question, and no detailed studies to characterize the peculiar magnetic properties have yet been reported.

In this paper, we investigated the magnetic properties of the new $S$ = 1/2 hyper-honeycomb lattice compound CCCO by magnetic susceptibility and nuclear magnetic resonance (NMR) measurements. NMR being a powerful local probe sheds light on the static and dynamic magnetic properties via spectra and nuclear spin-lattice relaxation rate (1/$T_{1}$) measurements, respectively, from a microscopic point of view. From the $T$ dependence of the NMR spectrum, we found that the spin susceptibility does not vanish at 50 mK, suggesting a non-spin-singlet ground state in CCCO. The $T$ dependence of $1/T_{1}$ indicates a slowing down of Cu spin fluctuations with decreasing $T$ down to 100 mK without showing any signature of long-range magnetic ordering. Our NMR data suggest that the magnetic behaviors of CCCO are characterized as an AFM chain system where the Cu spins fluctuate very slowly in the ground state.

Rod-shaped single crystals of CCCO were synthesized by the method described elsewhere zhang12a . The size of each crystal is $\leq 1\times 0.3\times 0.3$ mm³ and we used several crystals which are aligned along the rod direction (the $b$ axis). Magnetic susceptibility $\chi$ measurements were carried out using a Magnetic Property Measurement System (MPMS) from Quantum Design, Inc., in the $T$ range 1.8–350 K under a magnetic field $H$ = 0.1 T for two different magnetic field directions: $H$ parallel to the $b$ axis ($\chi_{||b}$) and perpendicular to the $b$ axis ($\chi_{\perp b}$). NMR measurements down to 50 mK were performed on ¹H ($I$ = 1/2, $\gamma_{\rm N}/2\pi$ = 42.5774 MHz/T) using a homemade phase-coherent spin-echo pulse spectrometer and an Oxford Kelvinox dilution refrigerator. $H$ is applied perpendicular to the rod direction (i.e., $H$$\perp$$b$) for NMR measurements. ¹H NMR spectra were obtained by Fourier transform of the Hahn spin-echo signals at fixed magnetic fields. The ¹H spin-lattice relaxation rates (1/$T_{\rm 1}$) were measured with a saturation recovery method. The recoveries of longitudinal magnetization display stretched exponential behavior due to distributions of $T_{1}$ values. Therefore, $1/T_{1}$ at each $T$ is determined by fitting the nuclear magnetization $M$ versus time $t$ dependence after saturation using the stretched-exponential function $1-$$M$($t$)/$M({\infty})$ = exp[$-(t/T_{\rm 1})^{\rm\beta}$]. Here $M$($t$) and $M({\infty})$ are the nuclear magnetization at time $t$ after saturation and the equilibrium nuclear magnetization at time $t$ $\rightarrow$ $\infty$, respectively. The $T$ and $H$ dependences of $\beta$ are shown in the Supplemental Material SI . The large distribution in $T_{1}$ seen in the reduction of the $\beta$ values is observed at low temperatures below $\sim$ 1 K where slow spin dynamics has been observed as discussed below.

Shown in Fig. 1 is the $T$ dependence of the anisotropic $\chi_{||b}$ and $\chi_{\perp b}$ in the $T$ range 1.8 $\leq$ T $\leq$ 350 K. $\chi(T)$ for both magnetic field directions exhibit broad humps around 170–190 K which has been attributed to a structural anomaly zhang12a ; CuOx . The Curie-like increases in $\chi(T)$ at low temperatures are not intrinsic and most likely originate from impurities, as we do not see any corresponding effects on NMR spectrum described below. The solid lines are corrected $\chi(T)$ obtained by subtracting the impurity contributions. Broad maxima were observed at $T_{\rm max}\simeq$ 30–35 K, one of the characteristic properties of low-dimensional antiferromagnets originating from short-range order of spins. The average value of the Cu-Cu exchange interaction $J$ is estimated to be $J\sim k_{\rm B}T_{\rm max}/0.7\simeq$ 43–50 K using a honeycomb lattice model Johnston1997 . If we use an antiferromagnetic chain model as pointed out below, a similar value of $J\sim k_{\rm B}T_{\rm max}/0.641\simeq$ 47–54 K is obtained Johnston1997 , indicating the average value of $J$ can be consider to be of the order of $J$ $\sim$ 50 K. The $T$ dependence of $\chi(T)$ of the single crystal is consistent with that of polycrystalline sample reported in Ref. CuOx .

Figures 2(a) and 2(b) show the typical $T$ dependencies of ¹H-NMR spectra of CCCO at $H$ = 4.72036 T for $T$ = 0.5–275 K and $H$ = 2.60667 T for the low-$T$ region $T$ = 0.05–1.2 K, respectively. A sharp single NMR line with a full width at half maximum (FWHM) of 34 kHz was observed around zero NMR shift position at 275 K. With decreasing temperature, spectra become slightly broader and, below 1.2 K a few shifted peaks with very small intensity were observed whose positions were nearly independent of $T$. Figure 2(c) shows the $T$ dependence of the FWHM measured at $H$ = 4.72036 T, together with the data at $H$ = 2.14 T and 2.60667 T. With decreasing $T$, the FWHM increases slightly to $\sim$ 50 kHz down to 100 K and shows nearly $T$-independent behavior down to $\sim$ 4 K. In the present compound, there are 32 protons in an ammonium cation, which would be located at slightly different distances from the magnetic ions and with different spatial orientations. Therefore, a powder-pattern-like NMR spectrum will be observed due to the distributions of the internal field at the proton sites, even though we used single crystals. In general, there are two possibilities for the origin of the internal field at the proton sites: classical dipolar and contact-hyperfine fields from the Cu spins. The classical dipolar fields yield a broadening (and/or shoulder) of the line but no net shift, while a net shift of the line can be produced by the contact hyperfine field due to the overlap of the $s$-electron wave function of a proton with the $d$-electron wave function through the oxygen and nitrogen ions. No NMR line with a clear net shift is observed above 1.2 K. This indicates that the broadening of the line is mainly attributed to the classical dipolar effects. In this case, the FWHM of the NMR line is expected to be proportional to $\chi$. Thus the increase of FWHM with decreasing $T$ from 300 K to 100 K can be attributed to the increase of $\chi$. The $T$-independent behavior of FWHM below 100 K could be due to the gradual structural transition around 150 K and also a disorder-order transition of one ethylene group on one of the ammonium cation at 165 K zhang12a ; CuOx , which produces the inhomogeneous broadening of the lines, masking the broad maximum around 35 K expected from the $T$ dependence of $\chi$.

With further decreasing $T$, the FWHM increases up to $\sim$ 100 kHz and levels off below 1 K. The FWHM at low $T$ is nearly independent of $H$ as shown in Fig. 2(d). The nearly $H$-independent behavior of FWHM at low $T$ indicates that the broadening of the spectrum arises from a nearly static internal field. The observation of a static internal field at nuclear sites depends on the fluctuation frequency of the spins. Whenever the fluctuation frequency is lower than NMR frequency, one may observe static internal fields at nucleus sites. On the other hand, the internal fields are time averaged to be zero when the fluctuation frequency is higher than the NMR frequency. The observation of the static internal field at the H sites therefore indicates that the fluctuation frequency of the Cu²⁺ spins is less than the NMR frequency of the order of MHz, evidencing the slowing down of the Cu²⁺ spins at low $T$, which is also consistent with the $T_{1}$ measurements described below. Thus, our experimental data clearly indicate that the Cu²⁺ spin moments exist at the lowest $T$ and the ground state of the compound is not a spin-singlet state, consistent with a gapless ground state.

To investigate the dynamical properties of the Cu²⁺ spins and the ground-state properties, we have performed proton spin-lattice relaxation rate 1/$T_{1}$ measurements in a wide $T$ range 0.1–275 K. Figure 3(a) shows the $T$ dependence of $1/T_{1}$ of ¹H NMR at various $H$. 1/$T_{1}$ exhibits a peak around 150 K, which is probably related to the freezing of the rotational motion of the C₂H₅ groups at the disorder-order transition at 165 K zhang12a ; CuOx . Below that temperature, 1/$T_{1}$ decreases and starts to increase at $\simeq$ 10 K, and then exhibits a peak at $\simeq$ 1 K. It is noted that a nearly $H$ independent behavior of 1/$T_{1}$ is observed at high $T$ while 1/$T_{1}$ strongly depends on $H$ at low $T$, especially below $\sim$ 1 K. As $H$ increases, the peak temperature of 1/$T_{1}$ shifts to higher $T$ and, at the same time, the height of the 1/$T_{1}$ peak becomes lower. Below $\sim$ 1 K, $T_{1}$ becomes longer with decreasing $T$ and the $T$ dependence is well reproduced by the thermal-activation behavior 1/$T_{1}$ $\propto$ exp$(-\Delta/k_{\rm B}T)$ as clearly seen in the semilogarithmic plot of 1/$T_{1}$ vs. $T$ in Fig. 3(b). Figure 3(c) shows the $H$ dependence of $\Delta/k_{\rm B}$ estimated from the slopes in Fig. 3(b). The values of $\Delta/k_{\rm B}$ seem to be proportional to $H$ and to follow the relation $\Delta/k_{\rm B}$ = 1.34$H$ K (here $H$ is in units of Tesla) as shown by the dashed line up to $\sim$ 4 T, but the relation does not reproduce the $H$ dependence of $\Delta/k_{\rm B}$ above 5 T. It is known that the magnitude of the field-induced gap is $\Delta/k_{\rm B}$ $\propto$ $H^{2/3}$ in AFM chains due to staggered $g$ tensors and/or Dzyalonshinskii-Moriya (DM) interactions Oshikawa1997 . In fact, since there is no inversion symmetry at the middle point between the Cu1 and Cu2 sites (see the inset of Fig. 1), DM interaction can affect the magnetic properties of the system. As shown by the red curve, the $H$ dependence of $\Delta/k_{\rm B}$ is roughly reproduced by the relation. This would be consistent with the prediction of the recent DFT calculations suggesting a weakly-coupled AFM Heisenberg chain system jacko21 . It should be emphasized that the magnitude of $\Delta/k_{\rm B}$ is estimated to be zero at $H$ = 0 from the $H$ dependence of $\Delta/k_{\rm B}$ in Fig. 3(c). This indicates no finite gap at zero magnetic field, in turn suggesting a gapless ground state in CCCO. It is noted that, although we cannot exclude a possible finite critical field for the $H$-induced gap, this will not change our conclusion.

It is important to point out that our $T_{1}$ data reveal a slowing down of Cu²⁺ spin fluctuations at low $T$. 1/$T_{1}$ is generally expressed by the Fourier transform of the time correlation function of the transverse component $h_{\pm}$ of the fluctuating local field at nuclear sites with respect to the nuclear Larmor frequency $\omega_{\rm N}$ as Abragam ,

Since the internal field at the H sites is mainly due to classical dipolar field, 1/$T_{1}$ is given by a sum of two contributions (1/$T_{1}$)_|| and (1/$T_{1}$)_⟂ due to the magnetic fluctuations parallel ($||$) and perpendicular ($\perp$) to the external field, respectively Moriya1956 . Assuming the time correlation function decays as exp(-${\it\Gamma}$$t$), 1/$T_{1}$ can be written by Hone1974 ; Maegawa1995 ; Giovannini1971 ; T1comment ,

where $\omega_{\rm e}$ is the electron Larmor frequency. $A_{i}$ and $\chi_{i}$ are the parameters related to the local fields at the H sites and the magnetic susceptibilities, respectively, for the $i$ (= $||$ and ${\perp}$) directions. Here the $\it{\Gamma}$ is the inverse of the correlation time of the fluctuation of the local fields at the H sites due to Cu²⁺ spins and is assumed to be isotropic for simplicity. As we discuss below, since the first term of Eq. 2 is considered to be dominant, we consider only the first term in the following.

When ${\it\Gamma}$ is independent of $T$, the $T$ dependence of 1/$T_{1}$ is simply proportional to $\chi_{||}T$. The solid curve in Fig. 3(a) shows the temperature dependence of $\chi_{\perp b}T$ where we used $\chi_{\perp b}$ for $\chi_{||}$ since the magnetic field is applied perpendicular to the $b$ axis. As can be seen in the figure, the $T$ dependence of 1/$T_{1}$ scales with that of $\chi_{\perp b}T$ above $\sim$20 K where 1/$T_{1}$ is nearly independent of $H$. The result indicates that the nuclear relaxations are induced by the paramagnetic fluctuations of Cu²⁺ spins whose frequency is much higher than the NMR frequency.

Below $\sim$20 K, on the other hand, the simple paramagnetic-fluctuation model cannot reproduce the experimental results. To analyze the $T$ and $H$ dependencies of 1/$T_{1}$ at low $T$ using Eq. 2, we re-plot the data by changing the vertical axis from 1/$T_{1}$ to 1/$T_{1}\chi_{\perp b}T$ as shown in Fig. 4(a), where $\chi_{\perp b}$ is assumed to be constant below 1 K since no clear change in the NMR shift and the FWHM were observed from the NMR spectra. According to Eq. 2, 1/${T_{1}T\chi_{\perp b}}$ is proportional to 1/${\it\Gamma}$ when ${\it\Gamma}$ $\gg$ $\omega_{\rm N}$ (fast-motion regime). This is actually observed above 20 K as discussed above. On the other hand, 1/${T_{1}T\chi_{\perp b}}$ is proportional to ${\it\Gamma}$/$\omega_{\rm N}^{2}$ in the case of ${\it\Gamma}$ $\ll$ $\omega_{\rm N}$ (slow-motion regime) where 1/$T_{1}$ is expected to depend inversely on the square of the $H$ since $\omega_{\rm N}$ = $\gamma_{\rm N}H$. When ${\it\Gamma}$ = $\omega_{\rm N}$, 1/${T_{1}T\chi_{\perp b}}$ reaches a maximum value.

Assuming $A$ = 1.9 $\times$ 10¹² rad$\cdot$mol/cm³/K/s² and ${\it\Gamma}$ = ${\it\Gamma}_{0}$exp($-\Delta/k_{\rm B}T)$ where we used ${\it\Gamma}_{0}$ $\sim$ 1.0 $\times$ 10¹⁰ Hz which slightly depends on $H$, the experimental results are well reproduced by Eq. 2, as shown in Fig. 4(a) by solid curves for different $H$. The dependence of the peak height of 1/${T_{1}T\chi_{\perp b}}$ on the NMR resonance frequency (i.e., $H$) is also well reproduced by the model as shown by the solid curve in the inset of Fig. 4(a). These results indicate that the peak observed in the $T$ dependence of 1/${T_{1}T\chi_{\perp b}}$ originates from a crossover between the fast-motion and the slow-motion regimes, whereby the fluctuation frequency of the Cu²⁺ spins below the peak temperature is less than the NMR frequency range which is of the order of MHz. This is consistent with the observation of the nearly $H$-independent broadening of NMR spectrum below 1 K. If we consider that the second term of Eq. 2 is relevant for the peaks of 1/$T_{1}$, ${\it{\Gamma}}$ at the peak position is the electron Larmor frequency, much higher than NMR frequency, which will not produce the broadening of the NMR spectrum observed below $\sim$1 K. These results suggest that 1/$T_{1}$ is mainly determined by the first term as pointed out, and thus we consider only the first term in our analysis.

To derive the $T$ dependence of the fluctuation frequency for the Cu²⁺ spins in a wide $T$ region, we extract the $T$ dependence of ${\it\Gamma}$ from the 1/${T_{1}T\chi_{\perp b}}$ data, assuming Eq. 2 is valid for all $T$ regions. The estimated $T$ dependences of ${\it\Gamma}$ for different $H$ are shown in Fig. 4(b). ${\it\Gamma}$ shows a thermally-activated behavior at low $T$ and is almost constant with ${\it\Gamma}$ $\sim$ 10¹⁰ Hz at high $T$. At $T$ below $\sim$ 1 K, the Cu²⁺ spins fluctuate with low frequencies, which is less than the NMR frequency of the order of MHz. No loss of NMR signal intensity and the absence of critical slowing down rule out the possibility of a spin-glass phase down to 50 mK in CCCO. This is further supported by the absence of a critical divergence of 1/$T_{1}$ or a cusp structure in 1/$T_{1}$ generally observed in a spin-frozen state Furukawa2015 .

In summary, we have carried out magnetic susceptibility and ¹H NMR measurements in a wide $T$ range from 0.05 K to 350 K on the quantum spin-liquid candidate CCCO to investigate its magnetic properties, especially focusing on its ground-state magnetic properties. Although the average AFM exchange interaction is $J$ $\sim$ 50 K from the broad maxima in the $T$ dependence of $\chi$, no long-range magnetic ordering down to 50 mK has been observed by our NMR measurements, consistent with the $\mu$SR measurements CuOx . We found that the fluctuation frequency of the Cu²⁺ spins slows down less than the NMR frequency of the order of MHz at $T\leq$ 1 K. In addition, from the $T$ and $H$ dependence of 1/$T_{1}$, the application of $H$ is found to give rise to a spin-excitation gap whose $H$ dependence is well reproduced by $\Delta/k_{\rm B}$ $\propto$ $H^{2/3}$. Our NMR data therefore suggest that the spin dynamics of CCCO essentially is characterized as an AFM chain, not a spin-liquid state, as recently pointed out by DFT calculations jacko21 .

The research was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. Ames Laboratory is operated for the U.S. DOE by Iowa State University under Contract No. DE-AC02-07CH11358. C.D. and Y.N. were supported by NSF CAREER DMR-1944975.