Nature of the spin liquid ground state in a breathing kagome compound studied by NMR and series expansion

TL;DR

This study investigates the magnetic properties of a breathing kagome lattice in a vanadium oxyfluoride compound using NMR and series expansion, revealing a likely gapless spin liquid state with specific interaction ratios.

Contribution

It provides the first detailed NMR analysis of a breathing kagome compound, quantifying interaction ratios and suggesting a gapless spin liquid ground state.

Findings

The ratio of interactions within the breathing kagome plane is approximately 0.55.

The mean antiferromagnetic interaction is around 60 K.

The excitation spectrum is essentially gapless with a very small maximum gap.

Abstract

In the vanadium oxyfluoride compound (NH$_4$)$_2$[C$_7$H$_{14}$N][V$_7$O$_6$F$_{18}$] (DQVOF), the V$^{4+}$ (3d$^1$, $S=1/2$) ions realize a unique, highly frustrated breathing kagome lattice composed of alternately-sized, corner-sharing equilateral triangles. Here we present an $^{17}$O NMR study of DQVOF, which isolates the local susceptibility of the breathing kagome network. By a fit to series expansion we extract the ratio of the interactions within the breathing kagome plane, $J_\triangledown / J_\vartriangle = 0.55(4)$, and the mean antiferromagnetic interaction $\bar{J}=60(7)$~K. Spin lattice, $T_1$, measurements reveal an essentially gapless excitation spectrum with a maximum gap $\Delta / \bar{J}=0.007(7)$. Our study provides new impetus for further theoretical investigations in order to establish whether the gapless spin liquid behavior displayed by DQVOF is intrinsic to its breathing kagome lattice or whether it is due to perturbations to this model, such as a residual coupling of the V$^{4+}$ ions in the breathing kagome planes to the interlayer V$^{3+}$ ($S=1$) spins.

Figures

Tables

Table 1. Table 1: Parameters of the series expansion fits of the NMR local susceptibility (see text).

$\alpha$	$J_{△}\text{ (K)}$	$J_{▽}\text{ (K)}$	$J_{▽}/J_{△}$	$\overline{J}=\left(J_{▽}+J_{△}\right)/2\text{ (K)}$	$\Delta$ (K)	$A$	$C{\text{ (cm}}^3.{\text{mol}}^{-1}{\text{ (V}}^{4+}\text{))}$
0	84(1)	49(1)	0.58	67(2)	3(1)	1.31	-2.6.10-4
1	73(1)	37(1)	0.51	55(2)	—	1.06	-2.11.10-5
2	68(1)	37(1)	0.54	53(2)	—	1.02	1.92.10-5

Equations

$${\chi }_{kago}^{th}(T)=\frac{1}{H}{\left(\frac{\partial f}{\partial H}\right)}_T.$$

$$E=\frac{1}{ϵn_T}\sum _{i=1}^{n_T}{T}_i{\left(A{\chi }_{kago}^{th}(T_i)+C-{\chi }^{loc}(T_i)\right)}^2$$

$$\frac{M(t_c)}{M_{sat}}=1-\frac{1}{35}{e}^{-t_c/T_1}+\frac{8}{45}{e}^{-6t_c/T_1}+\frac{50}{63}{e}^{-15t_c/T_1}$$

Full text

Frustrated magnetism stands at the forefront of theoretical and experimental quantum matter research and has emerged as a vast playground in which to realize novel exotic states Balents (2010). One emblematic example is the antiferromagnetic Heisenberg model for $S=1/2$ spins on the kagome lattice (KAFM) made of corner-sharing triangles. Despite its seemingly simple nature, the $S=1/2$ KAFM continues to provide a rich source of novel physics Ran et al. (2007); Messio et al. (2012) and challenges the most advanced analytical and numerical methods Yan et al. (2011); Depenbrock et al. (2012); Liao et al. (2016). The complexity of this problem stems from the many ground state candidates - mostly quantum spin liquid phases for which there is no associated symmetry breaking of the Hamiltonian - which compete within a narrow range of energy. Even the existence of a gap in the excitation spectrum is still hotly debated Nakano and Sakai (2011); Iqbal et al. (2013, 2014); Liao et al. (2016). In parallel, advances in materials science have led to the identification of several spin liquid candidates among which herbertsmithite, ZnCu3(OH)6Cl2, realizes the closest experimental approximation to the KAFM Shores et al. (2005); Mendels and Bert (2011); Norman (2016). However, the exact nature of the ground state of herbertsmithite remains unclear. First, the presence of a sizeable amount of Cu2+ ions sitting out of the kagome layers pollutes the low temperature magnetic properties and renders it challenging to definitively conclude whether or not there is a spin gap of few kelvins in the excitation spectrum Olariu et al. (2008); Fu et al. (2015); Han et al. (2012). Second, the absence of a firm theoretical prediction for the ideal KAFM model does not allow for a deep quantitative analysis nor for an evaluation of how perturbations from the ideal KAFM Hamiltonian that inevitably exist in real materials, such as defects, anisotropic interactions or further nearest neighbor interactions, influence the nature of the ground state.

A modified version of the kagome lattice, the trimerized, or breathing kagome lattice Schaffer et al. (2017) named after the breathing pyrochlores Okamoto et al. (2013), may offer an alternative route for the study of quantum spin liquids. Importantly, it also consists of corner-sharing equilateral triangles and hence it retains the full degeneracy of the kagome lattice at the classical level. However the alternation of the interactions within the triangles pointing up, $J_{\vartriangle}$, and those pointing down, $J_{\triangledown}$, (see Fig 1, inset) extends the problem of the isotropic KAFM ($J_{\vartriangle}=J_{\triangledown}$) to a larger class of highly frustrated systems. The trimerization may then select more clearly one type of spin liquid for the ground state Schaffer et al. (2017). This would constitute a clear theoretical standpoint to investigate the spin liquid physics in the recently discovered vanadium oxyfluoride compound (NH4)2[C7H14N][V7O6F18] (DQVOF) Aidoudi et al. (2011) which features such a breathing kagome lattice and which is the focus of the present study.

The structure of DQVOF consists of magnetic trilayer well separated by non-magnetic quinuclidinium [C7H14N]+ cations. Each trilayer contains two V4+ (3d1, $S=1/2$) kagome layers separated by a triangular interlayer of V3+ (3d2, $S=1$) ions (see insets in Fig. 1). The V3+ ions are located above or below every other triangle of V4+ ions, which yields the two set of alternating equilateral triangles within the $S=1/2$ breathing kagome lattice. All the vanadium ions are coupled by superexchange via fluoride ions which leads to antiferromagnetic Curie-Weiss behavior at high temperature with a Weiss temperature $\theta=58(4)$ K. Due to the compression of the octahedral environment of the V4+, the $d_{xy}$ orbital of the V4+ is singly occupied 111The VOF5 octahedron is strongly distorted with a distance $d_{V^{4+}-O}=1.58$ Å much shorter than the distances $d_{V^{4+}-F}$ all in the range $1.95-2.15$ Å which lifts the $t_{2g}$ orbitals degeneracy and favors the $d_{xy}$ one Aidoudi et al. (2011).. This lies close to the kagome plane and, therefore, little coupling is expected in the perpendicular direction to the V3+ ions. Indeed, previous magnetization and heat capacity measurements Clark et al. (2013) revealed the paramagnetic-like response of the V3+ ions, which have a weak residual effective coupling $\sim\leavevmode\nobreak\ 1$ K . As such, the $S=1/2$ kagome layers appear to be magnetically isolated from one another. Muon spin relaxation experiments down to 20 mK revealed no sign of any on-site frozen moments Clark et al. (2013); Orain et al. (2014) as expected for a spin liquid ground state.

In this Letter, we report the 17O nuclear magnetic resonance (NMR) study of DQVOF which isolates the local magnetic properties of the V4+ breathing kagome layers. The comparison between the local susceptibility and our theoretical calculations based on series expansion allows us to determine the ratio of the interactions $J_{\triangledown}/J_{\vartriangle}\sim 0.55$ while the dynamical spin lattice relaxation ($T_{1}$) measurements reveal a gapless excitation spectrum.

A polycrystalline sample (84 mg) was synthesized as described in Aidoudi et al. (2011) with the addition of 50 $\mu$l of 90 % 17O enriched water in the ionic solution Griffin et al. (2012). The NMR spectra were recorded from 10 - 300 K in a fixed field of 7.5 T by sweeping the frequency, while the broader spectra below 10 K were measured with a fixed irradiating frequency by sweeping the field from 6.5 T to 7.15 T and integrating the spin echo.

The 17O NMR spectrum at 300 K (Fig 1) is a typical powder spectrum for an $I=5/2$ nuclear spin in an almost axially symmetric environment. It consists of one sharp central line, i.e. one oxygen site as expected from the crystal structure, in between two pairs of well defined quadrupolar singularities. The observation of one single central line together with the small distribution of the quadrupolar parameters 222From a powder spectrum simulation, we extract an average quadrupolar frequency $\nu_{Q}=370$ kHz with a small distribution $\Delta\nu_{Q}=10$ kHz and a maximal anisotropy of the quadrupolar tensor $\eta<0.05$. The shift tensor $\bar{\mathbf{K}}$ is almost isotropic with $K_{x,y}/K_{z}=0.87$, negligible in view of the low $T$ broadening. demonstrates the absence of any sizable disorder and, in particular, of any magnetic dilution of the $S=1/2$ kagome layers.

Upon cooling the sample (Fig 2), the NMR spectrum first shifts negatively away from the reference line down to approximately 30 K. At lower temperature, it shifts back towards the reference and broadens drastically. The NMR shift is related to the local spin susceptibility $\chi^{loc}$ through $K=\mathcal{A}\chi^{loc}(T)+K_{0}$ where $K_{0}$ is the $T$-independent shift including the orbital shift and $\mathcal{A}$ is the isotropic hyperfine coupling. Given that the oxide ions are directly bonded to the V4+ ions within the DQVOF structure at a distance of 1.58 Å and are far away from the other magnetic centers, there is little doubt that the 17O nuclei are hyperfine coupled to those V4+ and thus that the lineshift predominatly probes the local susceptibility of the $S=1/2$ kagome layer. Note that, being a traceless tensor, the dipolar field arising from the polarization of the V3+ does not contribute to the lineshift in a powder sample but to the linewidth, yielding in particular the low $T$ broadening when the magnetization of the quasi-free V3+ increases drastically SI . A plot of the high temperature shift ($T>80$ K) versus a Curie-Weiss model for the V4+ kagome susceptibility SI gives $\mathcal{A}=-11.0(5)$ kOe.$\mu_{B}^{-1}$ and $K_{0}=1200(200)$ ppm. The local spin susceptibility $\chi^{loc}$ of the kagome layers depicted in Fig. 3 reveals a broad maximum at 30 K$\approx\theta/2$, in striking contrast to the monotonic variation of the macroscopic susceptibility (see inset, Fig. 3) which, unlike the local susceptibility measured by 17O NMR, encompasses the contribution of the quasi-free V3+ spins. The decrease of $\chi^{loc}$ at low temperatures provides evidence for the enhancement of the short-range antiferromagnetic spin correlations without long-range magnetic order which is in-keeping with the spin liquid description of the ground state Clark et al. (2013). The overall behavior of the local susceptibility of DQVOF is in fact very similar to that reported for herbertsmithite Fu et al. (2015).

We now turn to compare the experimentally determined $\chi^{loc}$ with the susceptibility $\chi^{th}_{kago}$ of the $S=1/2$ Heisenberg model on the breathing kagome lattice, calculated over the whole temperature range using the series expansion method described in Ref. Bernu and Lhuillier (2015); SI . In brief, this method relies on the extrapolation of the entropy $s(e,H)$ for an applied magnetic field, $H$, over the whole range of energies, $e$, starting from the high temperature series expansion of $s(T)$ and $e(T)$. The free energy $f=e-Ts$ is then computed at all $T$ and yields the susceptibility

[TABLE]

The extrapolation of $s(e,H)$ at low energy depends on the ad hoc choice of the type of behavior of the specific heat $C_{V}(T)$ at $T\rightarrow 0$. We have tried three cases as shown on Fig. 3: $i)$ gapped, hereafter noted $\alpha=0$ (red line), $ii)$ power laws : $C_{V}\propto T^{\alpha}$ with $\alpha=1$ (green line) and $2$ (blue line). The theoretical curves were fitted to the experimental data in order to minimize the fit error $E$ defined as

[TABLE]

where $n_{T}$ is the number of data points, $T_{i}$ the data temperatures, $\epsilon=0.0012$ an estimate of the experimental error on $T(\chi^{loc})^{2}$, $A\simeq 1$ and $C$ are corrections to the experimentally determined hyperfine coupling and $T$-independent shift, respectively. In view of the experimental data, we further imposed that $\chi^{th}_{kago}(T=0)=0$.

The fitting parameters obtained in this way are summarized in Table 1. We found fits of similarly high quality for the three possible values of $\alpha$ and it is thus impossible to conclude on the existence of a spin gap on the sole basis of the susceptibility data. In the gapped case ($\alpha=0$), the gap value refines to $\Delta=3(1)$ K $=0.05(2)\bar{J}$, in the range of the currently debated DMRG results for the isotropic kagome lattice Depenbrock et al. (2012); Nishimoto et al. (2013); He et al. (2016). Former analysis of the low temperature heat capacity of DQVOF, after subtraction of field dependent contributions, rather supports the linear $\alpha=1$ case Clark et al. (2013). Nonetheless, the various fits give a consistent determination of the breathing ratio $J_{\triangledown}/J_{\vartriangle}\simeq 0.55(4)$ providing firm ground for further numerical studies based on this ratio. Note that enforcing an isotropic model $J_{\triangledown}=J_{\vartriangle}$ yields unphysical parameters SI . Moreover, our attempts to fit the total SQUID susceptibility in the range 20 - 300 K by adding a simple Curie-Weiss contribution for the interlayer V3+ (Fig. 3, inset and SI ) gave a reasonably consistent value for the ratio of the interactions $J_{\triangledown}/J_{\vartriangle}\simeq 0.45(12)$.

In order to investigate the possible presence of a spin gap in DQVOF more precisely, we have performed NMR spin lattice ($T_{1}$) relaxation measurements which probe the spin excitation spectrum through $1/T_{1}\propto T\chi^{\prime\prime}(\omega_{n},T)/\omega_{n}$ where $\omega_{n}/2\pi=\leavevmode\nobreak\ ^{17}\gamma H_{0}/2\pi$ is the NMR frequency. The $1/T_{1}$ relaxation rates were measured at the top of the central line using a saturating pulse sequence $\pi/2$-$t_{c}$-$\pi/2$-$\tau$-$\pi$. The recovery curves (Fig. 4, inset) could be fitted over the whole $T$ range using the expression

[TABLE]

corresponding to the irradiation of the central line of an $I=5/2$ nucleus Suter et al. (1998). Hence a single $T_{1}$ is determined unambiguously at all $T$ and the absence of distribution of $T_{1}$ down to the base temperature points to a homogeneous ground state. Note that the V3+ ions are fully polarized at low $T$ in the 7 T applied field Clark et al. (2013) and therefore do not pollute the measurements of the kagome spin relaxation. The thermal evolution of $T_{1}$ is remarkably featureless, showing a weak power law increase $T^{\sim 0.3}$ over two decades in temperatures encompassing the energy scale of the interactions $\bar{J}$. At 1.2 K the $1/T_{1}$ value falls short of this behavior and may suggest a change towards slower dynamics. To capture this possible change of regime we have fitted the $1/T_{1}$ data over the whole $T$ range to the phenomenological function $1/T_{1}\propto T^{\beta}\exp(-\Delta/T)$, which accounts for a crossover from a pure power law behavior at high temperature to a gapped behavior at low $T$. We obtained $\Delta=0.4(4)$ K$\approx 0.007(7)\leavevmode\nobreak\ \bar{J}$ and $\beta=0.30(3)$. The vanishingly small value of $\Delta$, well below the estimates of the gap 0.05 J-0.13 J for gapped spin liquids in the isotropic model, points to an essentially gapless excitation spectrum for the breathing kagome lattice within DQVOF, in line with the linear $T$-dependence of the heat capacity suggesting gapless fermionic spinon excitations Clark et al. (2013).

We now consider the effect of the breathing of the kagome lattice on the spin gap issue. For nearly isolated spin clusters, one expects a large spin gap, $\sim J_{\vartriangle}$, from the discrete energy spectrum as observed for instance in LiInCr4O8 that features well decoupled tetrahedral clusters Okamoto et al. (2013). Increasing the intercluster interaction lifts the degeneracy of the individual ground states. A small gap, $\ll J_{\vartriangle}$, may then potentially open as suggested in early studies in the strong coupling regime ($J_{\triangledown}\ll J_{\vartriangle}$) Mila (1998); Mambrini and Mila (2000); Zhitomirsky (2005). A recent variational Monte Carlo study suggested a gapped state also for $0.5<J_{\triangledown}/J_{\vartriangle}<0.8$ Schaffer et al. (2017). The evidence for a gapless ($\Delta<0.007(7)\bar{J}$) spin liquid in DQVOF clearly calls for further theoretical investigation into the breathing KAFM including the calculation of the spin gap value, if any. Understanding how the spin liquid physics evolves with breathing and eventually connects to the isotropic kagome case is per se an interesting theoretical issue. Furthermore, in DQVOF, the presence of V3+ interlayer ions is likely the dominant perturbation to be considered beyond the breathing kagome model. This situation bears some interesting similarities with the widely-debated Cu2+ interlayer defects in herbertsmithite. The relevant energy scale of this perturbation is the coupling between the interlayer spins and the spins in the kagome planes. In both cases this interaction is likely to be weak but difficult to estimate. Earlier studies suggested an effective $\sim$1 K interaction Clark et al. (2013); Orain et al. (2014) which likely involves the kagome layer spins since no obvious exchange path exists between the V3+ ions. We found, in particular, that their contribution to the low temperature susceptibility cannot be described satisfactorily by a simple Curie-Weiss model SI . The effect of such ’dangling’ spins has not been widely investigated so far, but may yield rich physics encompassing the Kondo effect Kolezhuk et al. (2006); Gomilsek et al. (2016) or subtle correlation effects through the kagome layer Han et al. (2016).

In conclusion, using NMR techniques we have determined the local $S=1/2$ V4+ susceptibility of DQVOF, which can be accounted for by series expansion for the breathing kagome model. The determination of the ratio of the interactions within the breathing kagome lattice, $J_{\triangledown}/J_{\vartriangle}=0.55(4)$, will allow for further theoretical investigations into the effects of the breathing of the $S=1/2$ KAFM lattice that are specific to DQVOF. It is hoped that this in turn will shed light on our experimental observation of a gapless spin liquid ground state. The breathing of the kagome lattice appears to be an interesting tuning parameter that may be used to control the spin liquid physics and one that warrants further exploration. Investigating new anisotropic kagome compounds with different layered structures is also a promising experimental route to furthering our understanding of the rich physics of the $S=1/2$ KAFM Clark et al. (2015).