Unconventional spin dynamics in the honeycomb-lattice material $\alpha$-RuCl$_3$: high-field ESR studies

TL;DR

This study uses high-field ESR to explore the complex spin dynamics in $ ext{RuCl}_3$, revealing unconventional multiparticle excitations and a field-induced energy gap consistent with theoretical predictions.

Contribution

It provides new experimental evidence of multiparticle spin excitations and detailed high-field ESR spectra in $ ext{RuCl}_3$, advancing understanding of Kitaev physics.

Findings

Detection of two antiferromagnetic resonance modes in the ordered phase

Observation of a rich excitation spectrum in the quantum paramagnetic phase

Consistency of ESR data with the field-induced energy gap from thermodynamic measurements

Abstract

We present high-field electron spin resonance (ESR) studies of the honeycomb-lattice material $\alpha$-RuCl$_3$, a prime candidate to exhibit Kitaev physics. Two modes of antiferromagnetic resonance were detected in the zigzag ordered phase, with magnetic field applied in the $ab$ plane. A very rich excitation spectrum was observed in the field-induced quantum paramagnetic phase. The obtained data are compared with results of recent numerical calculations, strongly suggesting a very unconventional multiparticle character of the spin dynamics in $\alpha$-RuCl$_3$. The frequency-field diagram of the lowest-energy ESR mode is found consistent with the behavior of the field-induced energy gap, revealed by thermodynamic measurements.

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Full text

Spin systems with honeycomb structures have recently attracted a great deal of attention, both theoretically and experimentally. It was proposed that some of such systems can be experimental realizations of the Kitaev-Heisenberg model Kitaev , which encompasses a variety of possible magnetic ground states (from a conventional Néel order to a quantum spin liquid) and emergent fractional excitations (e.g., Majorana fermions and gauge fluxes) Chaloupka ; Yadav ; Petrova ; Baskaran ; Knolle ; Janssen ; Do . An essential peculiarity of this model is the presence of anisotropic bond-dependent interactions, defined in the Hamiltonian (Eq. 1) by the Kitaev parameter $K$:

[TABLE]

where $S_{i}$ and $S_{j}$ are spin-1/2 operators at sites $i$ and $j$, respectively, $J$ is the Heisenberg exchange interaction, $m=x,~{}y,~{}z$ label the three different links of the lattice, and $h$ is the uniform magnetic field. The Kitaev physics is thought to be realized in Ir-based magnets (such as $A_{2}$IrO3, $A=$ Na or Li Chaloupka ; Jackeli ; Singh ; Takayama ), where, due to a strong spin-orbital interaction, the multiorbital $5d$ $t_{2g}$ state can be mapped into a single orbital state with a pseudospin $j_{eff}$ = 1/2.

Recently, the honeycomb-lattice material $\alpha$-RuCl3 [Fig. 1(a)] has been proposed as another promising candidate to exhibit Kitaev physics. The local cubic symmetry of $\alpha$-RuCl3 is almost perfect, in contrast to the iridates. As revealed experimentally Kubota ; Johnson , the magnetic susceptibility and magnetization of $\alpha$-RuCl3 are very anisotropic, evident of the low-spin state of Ru3+. Low-temperature neutron scattering measurements Banerjee ; Ran suggested a collinear zigzag-ordered magnetic structure, which is one of the magnetic states predicted by the Kitaev-Heisenberg model. Magnetic field applied in the $ab$ plane suppresses the long-range magnetic order, so that above the critical field $H_{c}\approx 7$ T the system is in a quantum paramagnetic phase Kubota ; Johnson ; Leahy . At about 23 T the system undergoes the transition into the magnetically saturated phase Johnson ; Rem_FSPS . One exciting property of the quantum paramagnetic phase is the presence of a field-induced energy gap, revealed experimentally by means of nuclear magnetic resonance and heat-transport measurements Baek ; Sears ; Hentric ; Wolter .

In this work, we present results of systematic high-frequency electron spin resonance (ESR) studies of $\alpha$-RuCl3 in magnetic fields up to 16 T, allowing us to gain a deeper insight into the nature and peculiarities of the spin dynamics in this material across different phases of its phase diagram.

Single-crystal $\alpha$-RuCl3 samples with typical sizes of 3x3x0.5 mm3 were prepared using a vapor transport technique starting from pure RuCl3 powder Banerjee . The samples were characterized using standard thermodynamic techniques; the obtained specific-heat data are consistent with published results Kubota , exhibiting a sharp peak at $T_{N}\simeq 7.5$ K and the onset of a broad anomaly near 14 K (the latter is attributed to the presence of stacking faults Johnson ). The ESR measurements were performed employing a 16 T transmission-type ESR spectrometer, similar to that described in Ref. Zvyagin_INSR . In our experiments, a set of backward-wave oscillators, Gunn diodes, and VDI microwave sources were used, allowing us to probe magnetic excitations in this material in the very broad quasi-continuously covered frequency range from ca 50 GHz to 1.2 THz. The experiments were done in the Voigt configuration with magnetic field applied in the $ab$ plane at temperatures down to 1.4 K.

A very rich excitation spectrum was observed at a temperature of 1.4 K (Fig. 2), revealing the presence of six absorption lines: modes A and B were detected in the low-field zigzag-ordered phase, while modes C, D, E, and F in the field-induced quantum paramagnetic phase.

Angular dependences of resonance fields for modes C, D, and E (measured at a frequency of 1119 GHz, $T=1.4$ K) are shown in Fig. 3. The experiment reveals the 60∘ periodicity of ESR fields, as expected for a honeycomb structure. The angles $0^{\circ}$ and $30^{\circ}$ correspond to [110] and [100] direction, respectively. We would like to stress the importance of these measurements, allowing us to confirm the very high, twin-free, quality of the single-crystalline samples we used.

The frequency-field diagrams of ESR excitations for $H\parallel[110]$ are shown in Fig. 4. As mentioned, two gapped ESR modes, A and B, were observed below $H_{c}$, where the system is in the zigzag-ordered state. The intensities of both modes decrease with increasing temperature, and at about $T_{N}\simeq 7.5$ K both resonance lines vanish (Fig. 5), evidenced that the detected ESR modes are indeed the modes of antiferromagnetic resonance (AFMR) in the long-range magnetically ordered zigzgag phase. The extrapolation of the frequency-field dependences of modes A and B to zero field rem_1 revealed gaps $\Delta_{A}=620$ GHz (which corresponds to 2.56 meV) and $\Delta_{B}=790$ GHz (3.27 meV), respectively [Fig. 4]. The gap $\Delta_{A}$ GHz is consistent with results of inelastic neutron scattering at the $\Gamma$ point Banerjee_2 , time-domain THz spectroscopy Little measurements (shown in Fig. 4 by a gray circle) and calculations Winter , providing clear evidence of magnetic excitations at the center of the Brillouin zone. It is important to mention that due to absence of the inversion symmetry on the second-nearest-neighbour bonds, the Dzyaloshinskii-Moria (DM) term in $\alpha$-RuCl3 is allowed, allowing in its turn, ESR transitions at the K point (Fig. 1). Such excitations at the Brillouin-zone boundaries were observed in a number of multisublattice antiferromagnets (see, e.g., Bar ; Sakai ; Zhe_BSCO ) and known as $exchange$ modes. Thus, we speculate that mode B can be interpreted as such an exchange AFMR mode. Both AFMR branches demonstrate pronounced softening with increasing field. It is noticeable, that the lowest-energy observed AFMR gap, corresponding to magnetic excitations at the $\Gamma$ point, remains open at $H_{c}$. The presence of additional anisotropic terms (such as, e.g., the staggered DM interaction) can be a reason of the observed phenomenon Zvyagin_CUP .

Now we would like to focus on the high-field spin dynamics in $\alpha$-RuCl3. Above $H_{c}\approx 7$ T, the long-range zigzag order is suppressed, and the system is in the magnetically disordered but strongly correlated quantum paramagnetic phase. There, four modes were observed. The corresponding frequency-field diagrams of magnetic excitations for $H\parallel[110]$ are shown in Fig. 4. In our experiments, the most intensive mode C can be detected at temperatures up to $\sim 20$ K (Fig. 2). Mode D is less intensive and was observed at temperatures up to $\sim 15$ K. Modes E and F are relatively week, but still can be detected at lowest available temperature, 1.4 K.

Using exact diagonalization (ED) calculations on a 22-spin cluster for an extended Kitaev-Heisenberg model, Yadav et al. Yadav predicted the presence of a field-induced gapped quantum spin liquid state. In addition to the Kitaev coupling $K=-5.6$ meV, in-plane $g$ factor $g_{ab}=2.51$, and isotropic Heisenberg exchange interactions between nearest-, second-nearest-, and third-nearest-neighbor sites, $J_{1}=1.2$ meV, $J_{2}=J_{3}=0.25$ meV, the model includes nearest-neighbor symmetric anisotropic exchange constants $\Gamma_{xy}=-1.2$ meV and $\Gamma_{zx}=-\Gamma_{yz}=-0.7$ meV. For the given set of parameters, a spin liquid state was predicted to exist in $\alpha$-RuCl3 between approximately 11.5 and 14 T Yadav . Notably, the crossover from the spin-liquid to a spin-polarized phase should be accompanied by a pronounced dip in the excitation energy at about 15 T Yadav . No indication of such a dip in magnetic fields up to 16 T (at least for the chosen direction of the applied magnetic field) has been reveled in our experiments.

Baek et al.  Baek performed ED calculations for the regular Kitaev-Heisenberg model on a 24-spin cluster, without symmetric anisotropic coupling and assuming $K=-10.0$ meV, $g_{ab}=2.4$, $J_{1}=2.0$ meV, $J_{2}=J_{3}=0.5$ meV. The calculations were done for intensities integrated over a broad momentum range and revealed a very rich excitation spectrum.

Recently, Winter et al. Winter_2 have reported another ED study of $\alpha$-RuCl3 for magnetic excitations at the $\Gamma$ point (which is the most common case for ESR) and including symmetric anisotropic coupling. They considered a simplified $C_{3}$-symmetric four-parameter model, assuming $J_{1}=-0.5$ meV, $K_{1}=-5.0$ meV, $\Gamma_{1}=2.5$ meV, $J_{3}=0.5$ meV. For fields $H>H_{c}$ the calculations showed a large redistribution of spectral weight, which can be attributed to the anisotropic frustration of the considered model. Comparison of the calculation results for magnetic field $H$ applied along the $a$ axis (which corresponds to the [110] axis in our experiments) with the ESR data revealed a very good qualitative agreement. The calculation results are shown as color scale together with the experimental data in Fig. 4. A particular good agreement was obtained for the most intensive ESR mode C, suggesting that this mode corresponds to excitations at the $\Gamma$ point. The observation of a number of ESR modes, not accounted by the calculations Winter_2 for magnetic excitations at the $\Gamma$ point, potentially suggests the presence of exchange ESR modes (see the discussion above). The corresponding calculations are in progress Winter_DM .

Numerical calculations Baek ; Winter_2 above $H_{c}$ predicted a rather complex excitation spectrum, consisting of a number of modes, whose activation energy increases with increasing field. A noticeable high-field property of the detected modes C and F is their unusually large slope $g_{ab}\mu_{B}\Delta S\approx 0.27$ meV/T (it is interesting, that the field dependence of mode F, the lowest mode observed in our ESR study in the $H>H_{c}$ region, matches that of the gap, extracted from the heat-transport experiments Hentric ). The remarkably large slope might imply the presence of ESR transitions with $\Delta S\approx 2$ (contrary to $\Delta S=1$, expected for elementary one-particle excitations in simple $S=1/2$ systems). This observation (together with results of the ED calculations, evident of a number of excitation modes, split off from the higher-energy continuum), suggests that the spin dynamics of $\alpha$-RuCl3 has an emergent multiparticle nature. To the best of our knowledge, such unconventional discrete (bound-state-like) multiparticle excitation spectrum has never been previously observed in magnetic systems with a honeycomb lattice. One can speculate that strong ferromagnetic Kitaev coupling $K$ may facilitate the formation of two-magnon bound states that are strongly split down from the two-magnon continuum. However, at present stage, it is unclear whether observed multiparticle excitations correspond to bound magnons or to bound Majorana spinons (and, in the latter case, whether the spinons are confined or simply bound). We hope that our data will stimulate further theoretical studies of the unusual spin dynamics in $\alpha$-RuCl3, in particular, in its high-field phase.

In conclusion, we performed comprehensive high-field ESR studies of $\alpha$-RuCl3, an anisotropic spin system with a honeycomb structure that is considered a top candidate for exhibiting Kitaev’s spin-liquid physics. Our experiments revealed the presence of two soft AFMR modes in the zigzag-ordered phase and uncovered the rather complex spin dynamics in the field-induced quantum paramagnetic state, characterized by emergent multiparticle excitations. Our observations can have a broader impact, suggesting that honeycomb-lattice magnets might serve as an excellent playground to study unconventional many-body quantum processes in condensed matter, including, e.g., field-induced condensation of bound states (with a potential realization of the spin-nematic order), Efimov effect, etc.

Note: Upon finalizing this manuscript, we became aware of two other high-field spectroscopy studies of $\alpha$-RuCl3, by Wang et al. Zhe and Wellm Wellm . Apart from the low-frequency AFMR mode and a number of excitations in the field-induced disordered phase (partially similar to that observed by us), a signature of a broad excitation continuum, an indication of the unconventional multiparticle spin dynamics in $\alpha$-RuCl3, was revealed.

This work was partially supported by Deutsche Forschungsgemeinschaft (project ZV 6/2-2) and by the HLD at HZDR, member of the European Magnetic Field Laboratory (EMFL). A. Banerjee and S. E. Nagler were supported by the Division of Scientific User Facilities, Basic Energy Sciences US DOE, P. Lampen-Kelley and D. G. Mandrus by the Gordon and Betty Moore Foundations EPiQS Initiative through Grant GBMF4416, J.-Q. Yan and C. A. Bridges by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division. We acknowledge discussions with S. M. Winter, K. Riedl, R. Valentí, M. Zhitomirsky, G. Jackeli, and S. Nishimoto. We thank Ch. Balz for his help orienting the samples.