Field evolution of magnons in -RuCl by high-resolution polarized terahertz spectroscopy
The Kitaev quantum spin liquid (KSL) is a theoretically predicted state of matter whose fractionalized quasiparticles are distinct from bosonic magnons, the fundamental excitation in ordered magnets. The layered honeycomb antiferromagnet -RuCl is a KSL candidate material, as it can be driven to a magnetically disordered phase by application of an in-plane magnetic field, with T. The spectrum of magnetic fluctuations at zero wave vector ( = 0) in the antiferromagnetic state is of central importance to understanding the proximity of this system to a fractionalized spin liquid. Here we report a detailed characterization of this spectrum by high-resolution time-domain terahertz (THz) spectroscopy. We observe two sharp magnon resonances whose frequencies and amplitudes exhibit a discontinuity as a function of applied magnetic field, as well as two broader peaks at higher energy. Motivated by the existence of well-defined spin wave resonances, we compare these results to calculations based on linear spin wave theory (LSWT). Below the Néel temperature, we find that LSWT can account for these essential features of the spectra when a -breaking distortion of the honeycomb lattice and the presence of structural domains are taken into account. We conclude that in the antiferromagnetic state the low energy = 0 spectrum in -RuCl is dominated by well-defined magnons, even when approaching , that are well-described by a semi-classical, non-interacting theory.
The quantum spin liquid (QSL) is an exotic phase of matter characterized by a disordered yet highly entangled ground state. Geometrically frustrated magnets with, for example, a triangular arrangement of spins have been predicted to host such states. Another promising route to a QSL is the Kitaev honeycomb, which consists of spin-1/2 particles arranged on a hexagonal lattice. Kitaev (2003, 2006) In this model, anisotropic Ising-like exchange interactions between nearest neighbors give rise to frustration. The ground state is a gapless Z spin liquid, with excitations taking the form of itinerant Majorana quasiparticles and static fluxes.
The Kitaev honeycomb is of recent experimental interest, as the anisotropic interactions characteristic of the model can manifest in real materials, Jackeli and Khaliullin (2009); Chaloupka et al. (2010) in particular transition metal compounds with strong spin-orbit coupling (SOC) such as the Na and Li iridates Williams et al. (2016); Biffin et al. (2014); Modic et al. (2014); Takayama et al. (2015) and -RuCl. Kim et al. (2015); Plumb et al. (2014) Despite the presence of a Kitaev term in the effective spin Hamiltonian, these materials order magnetically at low temperatures Liu et al. (2011); Choi et al. (2012); Chun et al. (2015); Biffin et al. (2014); Takayama et al. (2015); Sears et al. (2015) indicating that they host interactions beyond Kitaev exchange. Characterizing these interactions can help to navigate the rich phase diagrams of these materials, wherein one may approach a quantum-disordered state by applying external perturbations such as fields or chemical substitution. Yadav et al. (2016)
-RuCl has risen to prominence as a candidate Kitaev system, driven by the availability of single crystals suitable for inelastic neutron scattering (INS) Banerjee et al. (2016) and optical spectroscopy Little et al. (2017); Wang et al. (2017a); Shi et al. (2018), as well as the observation that magnetic order disappears in an in-plane magnetic field 7.5 T Sears et al. (2017); Baek et al. (2017); Zheng et al. (2017); Leahy et al. (2017); Banerjee et al. (2018). In this material, quasi-2D layers of Ru atoms surrounded by Cl octahedra are arranged on a honeycomb lattice. The combination of octahedral crystal field splitting, electron correlations, and SOC gives rise to a Mott-insulating state with a localized J = 1/2 moment on each Ru site. Kim et al. (2015) The quasi-2D layers are stacked and van der Waals coupled to form bulk -RuCl. Such layered magnets are of particular interest because they can be assembled and stacked with other 2D materials, forming heterostructures with potentially topological phases. Soumyanarayanan et al. (2016)
Spectroscopic probes such as INS, Raman scattering, and THz absorption have been employed to characterize magnetic fluctuations in -RuCl and test for the existence of, or proximity to, a QSL phase. Below T = 7 K, the ground state is a zigzag antiferromagnetic order Johnson et al. (2015), as shown in Fig. 1. (b). In the ordered phase and in zero applied magnetic field, INS measurements Banerjee et al. (2016, 2018) observed peaks consistent with magnons together with a continuum of scattering centered at = 0 -point) that was seen as well by Raman spectroscopy. Sandilands et al. (2015) This continuum persisted at fields above , as well at temperatures above at zero field, and was interpreted as a possible signature of Majorana fermions and Z vortices. To assess this it is of great interest to understand what aspects of the spectrum can be understood as conventional magnon or multi-magnon scattering. Subsequent THz absorption measurements showed that below T the majority of the -point spectral weight at low energies was accounted for by spin waves, and furthermore, that the contribution from a magnetic continuum did not grow with increasing magnetic field, up until . Little et al. (2017) THz and ESR measurements that explored showed that sharp resonances in the -point spectrum also exist in the high field regime.Ponomaryov et al. (2017); Wang et al. (2017a)
In addition to spectroscopic probes, the phase diagram of -RuCl has been explored extensively by thermodynamic and transport measurements, including specific heat, Kubota et al. (2015); Wolter et al. (2017); Sears et al. (2017) nuclear magnetic resonance, Baek et al. (2017) and thermal transport,Leahy et al. (2017); Hentrich et al. (2018); Yu et al. (2018) that indicate a transition to a gapped magnetically disordered state. However, varying interpretations of the nature of that state and its low-energy excitations leave the question of a transition to a QSL at or near unresolved. Recent experiments reporting a quantized thermal Hall effect Kasahara et al. (2018) for off-axis applied fields, a signature of chiral Majorana modes, suggest that a topological phase may exist in the vicinity of .
Precise measurement of the magnetic fluctuation spectrum as approaches can shed light on the nature of this transition. Time-domain THz spectroscopy (TDTS) probes these fluctuations with high sensitivity and energy resolution, yielding an absolute measurement of the imaginary part of the dynamic magnetic susceptibility in the energy range 0.1 to 1.7 THz, or 0.4 to 7.0 meV. Little et al. (2017) In this work we report THz absorption measurements that fully characterize the portion of associated with the antiferromagnetic state of -RuCl as a function of static field and THz probe field . We observe four resonances whose frequency and amplitude exhibit a complex dependence on applied field that depends strongly on the relative orientation of and . We use the absolute determination of provided by THz absorption to track the the dependence of the spin wave spectral weight on for and . These spectral weights are then compared with the static susceptibility, to determine the relative contributions of spin wave vs. continuum to the total weight of magnetic fluctuations at zero wave vector.
Motivated by the existence of sharp spin wave resonances with large spectral weight throughout the antiferromagnetic phase, we compare our experimental results with calculations based on linear spin wave theory (LSWT). Surprisingly, we find that LSWT can account for all the essential features of the spectra – the number of modes, their spectral weight and optical selection rules, the variation of resonant frequency with , and a discontinuity in mode frequency and amplitude at a low field of 1.5 T. Importantly, the contribution to the spectrum from two-magnon states is clearly identified. Achieving this description requires considering the -breaking of the honeycomb lattice and the resulting multi-domain structure, as well as a refinement of existing parameterizations of the effective spin Hamiltonian.
Definition of axes.
To guide the polarized TDTS measurements, the optical anisotropy of -RuCl samples was first characterized by measuring their transmitted THz electric field amplitude when rotated between crossed linear polarizers. Fig. 1(a) shows a typical room temperature scan of transmission as a function of angle of rotation about the optic axis. The nearly four-fold pattern, observed in all samples studied, indicates breaking of symmetry. This result is consistent with X-ray diffraction measurements that indicate a elongation of one of the Ru-Ru bonds and a monoclinic space group. Johnson et al. (2015); Cao et al. (2016) Fig. 1(b) depicts a Ru honeycomb layer that forms this structure, where , , and label the Ising axis of the Kitaev exchange term on the Ru-Ru bonds. An elongation in the direction of one the bonds (the one labeled by in the sketch) defines the axis of the monoclinic structure. The color of the atoms illustrates the zigzag antiferromagnetic order that arises below the Néel temperature (T).
The absence of nodes in the polar pattern in Fig. 1(a) indicates that the local symmetry is broken globally by the presence domains of the three equivalent orientations of monoclinic distortion, which are rotated 120 from one another. A single domain crystal would exhibit zero transmission for THz fields polarized parallel to the or axes, which is not seen in any of the samples we have studied. On the other hand, in a sample containing equal populations of three domains the optical anisotropy of each would be effectively canceled and the THz transmission between crossed polarizers would vanish for all angles. What we observe instead is the intermediate case, where unequal domain population gives rise to weak residual anisotropy. To confirm the presence of multiple domains, we performed scanning X-ray micro-Laue diffraction measurementsTamura et al. (2003) that indeed revealed the presence of all three domains with spatially varying populations as discussed in the supplementary material, section II. This multi-domain character, as we will show, is essential to understanding the THz absorption spectra in the zigzag state as a function of magnetic field.
Because of the low effective symmetry, the directions of minimum transmission in Fig. 1(a) do not coincide with the monoclinic axes of a single domain, although they will be close to those of a dominant domain. In this study, we reference our THz polarization and external magnetic field to the two directions of minimum transmission, which we label as and to distinguish them from the monoclinic axes of a single domain. We measure the absorption coefficient with the THz probe field in the honeycomb plane, , oriented parallel to and , and in both cases we compare measurements with in-plane parallel and perpendicular to .
Magneto-optical THz spectroscopy. The magnetic dipole contribution to that is associated with the presence of antiferromagnetic order can be isolated by subtracting spectra measured at T = 8 K, which is sufficiently above T such that magnons are no longer present, from spectra in the ordered phase at T = 4 K (see supplementary material). The residual spectrum omits any magnetic contribution that does not change when crossing T. Figs. 1(c) and (d) show differential (4 K - 8 K) absorption spectra, , for a sample of thickness mm for H parallel to . In the parallel () channel (Fig. 1(c)), a single magnon is observed at = 2.6 meV (0.62 THz) for = 0, which shifts to lower energy and broadens as the field is increased. The spectra measured with , shown in Fig. 1(d), are more complex in that the frequency and spectral weight appear to vary non-monotonically in field. In addition, two broader features, which we denote by , and , appear in the energy range 4-6 meV and become more strongly absorbing as the field is increased.
The evolution of the spectra with is greatly clarified by the color scale plots in Fig. 2, which illustrate the magnitude of in the plane. Panels (a), and (b) show spectra with , for parallel and perpendicular to , respectively. Panels (c), and (d) are the corresponding spectra for the configuration. Panels (e) and (f) show fits obtained by LSWT calculations discussed below.
We first note that the anisotropy with respect to rotation of the crystal by 90 is weak, that is, the pair of panels (a) and (b) share the same qualitative features as panels (c) and (d), with overall amplitude difference of only . As we discuss below, LSWT predicts a much larger anisotropy in the dynamic susceptibility between the two principal ( and ) axes of a single zigzag domain. We interpret the observed weak anisotropy to be further evidence for the presence of multiple domains. The width of the peaks remains relatively constant until around H 5 T at which point they start to broaden.Little et al. (2017) The broadening occurs more rapidly for as is apparent in Fig. 2 (a), where the magnon becomes diffuse approaching 7 T. This is an indication that for the system is close to the critical point and corrections to the spin wave expansion become relevant (see supplementary material, section VI).
A far stronger contrast is seen when comparing spectra with (panels (a) and (c)) to (panels (b) and (d)). For we observe a single mode that shifts to lower frequency with increasing , with the field-induced mode softening slightly more pronounced with . For the color plots show clearly that, rather than a single mode with a non-monotonic dependence of energy on field, there are in fact two distinct low energy modes. At there is a strong mode, = 2.6 meV, and a much weaker one, = 3.3 meV. We note in particular the 0.7 meV splitting between these modes, which informs our LSWT calculations. As increases the spectral weight of decreases rapidly and then shifts to for H T. Surprisingly, the total spectral weight at this crossover field is close to zero.
The absorption features centered at meV and meV at H = 4 T, grow with increasing and persist as approaches . An exact diagonalization study of -RuCl associated eigenstates in this energy range with a two-magnon continuum. Winter et al. (2018) Our results for using LSWT described in the next section (and shown in Fig. 4(f)) account for the field and polarization dependence of and , and confirm their origin as two-magnon excitations in the longitudinal response, that is, parallel to the zigzag wavevector.
The differential THz absorption is directly related to the imaginary part of the zero wave vector dynamic susceptibility ), that is,
where is the speed of light in -RuCl in the THz regime, which is determined independently (see supplementary material, section I). The thermodynamic sum rule, derived from the Kramers-Kronig relation, relates to the dc magnetic susceptibility, . With this sum rule, the contribution to from = 0 spin waves can be determined from the spectral weight of the spin wave peaks,
where the subscript denotes the component of susceptibility originating from spin wave resonances. By comparing to we can place an upper bound on the spectral weight not accounted for by spin waves, i.e., a magnetic continuum. Little et al. (2017)
We evaluate by fitting a Lorentzian function to the THz resonances (see supplementary material, section V). The resulting is plotted in Fig. 3, for each of the four configurations of and shown in Fig. 2. Also shown in Fig. 3 is as a function of magnetic field, which is defined by the change in magnetization resulting from a parallel to . Note that in the channel tracks as they both increase with increasing field. The difference , which is an upper bound on the spectral weight of a magnetic continuum, persists but does not increase as approaches . Finally, we note a small feature near 5.5 T in the parallel configuration for both the and curves, roughly consistent with a proposed intermediate phase in the T range. Banerjee et al. (2018)
The dependence on field of the spin wave spectral weight measured with is shown as well in Fig. 3, where it is seen to be strikingly different from the results for . In this configuration the spectral weight exhibits a deep minimum at 2 T for both the and directions, where it nearly vanishes. The field at which this minimum occurs is the same as the field at which the crossover from the to takes place in the THz spectra. In the following section, we explain how the main features of these data can be modeled using LSWT.
Spin Wave Theory.
The starting point for the LSWT calculations is the effective spin Hamiltonian,
where and denote summation over nearest neighbor and third neighbor bonds, respectively. Rau et al. (2014); Winter et al. (2016); Yadav et al. (2016); Wang et al. (2017b); Winter et al. (2017) is the Kitaev interaction, is the symmetric off-diagonal term and , are the nearest-neighbor and third neighbor Heisenberg couplings, respectively. The are bond labels (, , or ) as shown in Fig. 1 (a) and are the two remaining directions for each bond. Note that the magnetic field is expressed in spin-space components, for example, is expressed as and is .
The parameters in Eq. 3 are chosen such that the classical ground state has the observed zigzag antiferromagnetic order. We obtain the collective modes by expanding the Hamiltonian to quadratic order in the fluctuations about the ordered magnetic moment. Holstein and Primakoff (1940); Toth and Lake (2015); Colpa (1978) The spin wave theory is reliable when quantum (or thermal) fluctuations are small compared to the ordered moment, in which case the normal modes are non-interacting magnons. We obtain the theoretical THz absorption by computing the linear response of the magnons to an oscillating magnetic field (see supplementary material, section VI).
In the zigzag state, the unit cell of the honeycomb is enlarged to include four sites; as such there are four independent dispersing magnon modes. Of these, only two contribute to THz absorption, corresponding to the and modes discussed above. The two higher energy modes cannot be excited by the uniform in-plane THz field. This selection rule is exact, and is a result of a symmetry of the zigzag state, whereby two pairs of spins within the unit cell may be exchanged (see the supplementary material section VIII). Thus we do not associate the observed peaks at and with these modes.
To find appropriate values for the parameters in Eq. Field evolution of magnons in -RuCl by high-resolution polarized terahertz spectroscopy, we began with the representative values chosen by Winter et al. Winter et al. (2018, 2017) to model INS data, and adjust them to fit the energies of the modes seen by TDTS. The parameterization of Winter et al. leads to an accidental degeneracy of modes and . However, accounting for our observation that these modes are split by 0.7 meV at = 0 requires a significant refinement of the parameters of . In particular we find that fitting the spectra is accomplished by taking rather than . A representative fit to the energies of modes and as a function of using the parameter set (, , , ) = ( -0.35, -2.8, 2.4, 0.34) meV is shown as dots in Fig. 2 (f). We assume the same in-plane g-factor of 2.3 as used by Winter et al. Winter et al. (2018, 2017) The calculated mode energies are an excellent fit to the measured values.
In the following we show that the polarization selection rules predicted by LSWT account for the intricate mode-switching behavior observed at intermediate magnetic fields, shown in Fig. 2 (a-d). The crossover at T coincides with the disappearance of magnetic Bragg peaks corresponding to one of the three possible orientations of zigzag order on the honeycomb lattice. Sears et al. (2017); Banerjee et al. (2018) Previously, this effect was interpreted assuming that three degenerate zigzag orientations are present as domains. Sears et al. (2017) Within this picture, application of a magnetic field lifts the 3-fold degeneracy, driving energetically favored domains to grow at the expense of others. The possibility that the disappearance of magnetic Bragg peaks is related to reorientation of the ordered moment direction was also discussed. Banerjee et al. (2018) In the following, we show that the explanation of the THz mode crossover and deep minimum in spectral weight seen at 1.5 T requires the breaking of symmetry, even at . As mentioned previously, the breaking of occurs with a relatively small elongation of one of the three bond directions. We incorporate this distortion into the spin Hamiltonian by reducing the coupling constants , and for the “stretched” bond. Breaking symmetry in this manner lifts the degeneracy between the three possible zigzag wave vectors, ; the zigzag with parallel to the direction of its stretched bond (local monoclinic axis) is energetically favored, the two other orientations of related by rotation are degenerate and higher in energy. This zero-field splitting plays a key role in shaping the field dependence of the THz spectra, as discussed below.
Our scenario for the evolution of the spectra with magnetic field is illustrated in Fig. 4, which presents a table of the energetically preferred states and active modes for each domain, for values of below and above 1.5 T. We label each bond direction by , , or , depending on the orientation of its Kitaev interaction. The hexagons with , , and -stretched bonds shown in the bottom row of the table illustrate the spin order of the three domains at , where the spins are projected onto the plane. Our calculations show that application of a magnetic field favors zigzag orientations for which is largest. At a crossover field, , the energy gain exceeds the zero-field splitting. For the zigzag wave vector in all domains aligns with the direction selected by the magnetic field, while structural domains remain intact. The field-induced crossover is illustrated in Fig. 4 for the case where the applied magnetic field favors the domain shown in the left-hand column, in which the bonds are stretched. For the zigzag wave vector of the and domains will reorient to the of the -stretched domain. This process is analogous to the usual spin-flop transition in antiferromagnets, with the distinction that here the rotation involves both the direction of the moments and wave vector of the magnetic order.
The “-flop” crossover described above accounts naturally for the complex evolution of the THz absorption with applied field, when we take into account the polarization and relative spectral weight of and . As illustrated by the arrows inside the ellipses in Fig. 4, for the preferred zigzag order of a z-stretched domain (), is excited by and by . The polarization of these modes reflects an approximate symmetry with respect to exchange of x and y spin coordinates within the zigzag state. This symmetry is exact at zero field, and is explained in further detail in supplementary material, section VIII. Furthermore, our LSWT calculations predict that the spectral weight of is approximately a factor of six larger than that of (as indicated by the eccentricity of the ellipses). Thus, LSWT predicts strong optical anisotropy for a single structural domain. The fact that the measured THz absorption is nearly isotropic in plane follows from the presence of the three structural domains with comparable, though unequal, population.
The state of the system for is indicated by the lower row of ellipses in Fig. 4. In this regime, for all directions of the spectrum is dominated by the strong mode at 2.6 meV, although at 3.3 meV appears faintly as well. The upper row of ellipses shows the reorientation of the polarization that accompanies the -flop crossover at . With all the ellipses now aligned with the applied field, there is suddenly a strong dependence on the relative orientation of and ; couples only to while couples only to . This results in the mode-switching from to that is observed only in the channel. Figs. 2(e) and (f) show the evolution of the THz absorption spectra calculated with LSWT on the basis of the above model, which accurately reproduces the complex field and polarization dependent features of the experimental data.
In Fig. 5, we show that the multi-domain LSWT theory described above captures the curious deep minimum in spectral weight for at 1.5 T (expressed as ). The upper theoretical curve is the classical result, while the lower curve includes zero-point fluctuations of the spin 1/2 moments. The sudden reduction in spectral weight for occurs when the applied field aligns the Q of each domain, such that at , couples only to the weaker mode. Although the crossover predicted by the theory is sharp when compared with experiment, broadening of the -flop crossover is expected in the presence of structural disorder. We note that our scenario is consistent with the increase of the M point spin-wave intensity at 2 T observed in INS measurements. Banerjee et al. (2018)
Finally, we discuss the features and that are observed in the channel in the photon energy range meV (Figs. 2 (b) and (d)). These modes cannot be identified as single magnon excitations because of the exact symmetry discussed above. However, LSWT predicts absorption by a continuum of two-magnon states in precisely this energy range (Fig. 2(f)). A further prediction is that the two-magnon absorption takes place selectively for parallel to the ordered moment. As shown in Fig. 4, for the moments have flopped to an orientation that is nearly perpendicular to . Thus the two-magnon interpretation of and is consistent with the selection rule seen in the data, as these features appear for and are unobservable for .
Although the selection rules show unambiguously that and are two-magnon excitations, the details of the calculated field dependence (Fig. 2(f)) differ from the data. This is in contrast to the excellent agreement in the case of the single-magnon modes and . The most likely origin of this discrepancy is that while the single magnon modes are measured at = 0 the two-magnon absorption depends on the spin wave dispersion over the entire Brillouin zone. While our LSWT parameters reproduce the local minima at the M-points seen by INS, they do not reproduce the local minimum observed also at the -point Banerjee et al. (2018) (see supplementary material, section X). Indeed, all the theoretical models of this system studied to date do not reproduce this feature of the INS data Winter et al. (2018); Suzuki and Suga (2018) However we find that a -point minimum appears within LSWT when further interactions are added, for example second nearest-neighbor ferromagnetic coupling. Finding a spin Hamiltonian that describes all aspects of the single-magnon, two-magnon, and INS spectra is a goal for future research.
Conclusion and Summary
In summary, we used polarized time-domain THz spectroscopy to track the dispersion of magnons in -RuCl approaching the 7.5 T transition to a spin disordered state. We employed LSWT, including symmetry breaking distortions and multiple structural domains, to closely model the THz spectra and characterize the relevant selection rules. The unusual field dependence of the modes observed in the perpendicular configuration is the result of a -flop crossover, where at = 1.5 T the external field overcomes the anisotropy of the crystal to select the preferred ordering wave vector of the zigzag state. We expect this feature to be a general property of systems with nearly degenerate zigzag orders. Furthermore, the higher energy features which appear near 4 T are consistent with a continuum of two-magnon excitations. Such close agreement between the observed THz spectra and our model is striking in the sense that intricate features of the data can be described within the assumptions of LSWT, even approaching the critical field. Even so, the competing interactions in this system present a complicated problem and nature of the disordered state just above remains an open question. By resolving the evolution of the magnon excitations in the context of local symmetry breaking, we refine the corresponding spin Hamiltonian parameters in the ordered phase and provide valuable input for future work.
Crystal Synthesis. The sample studied was synthesized at Oak Ridge National Lab. Commercial-RuCl powder was purified to a mixture of -RuCl and -RuCl, and converted to 99.9 phase-pure -RuCl by annealing at 500C. Single crystals of -RuCl were grown using vapor transport at high temperature. The sample used for THz is roughly 5 mm x 8 mm in size and 1 mm thick. The index of refraction (n = 2.5 in the THz regime) and registration of optical and crystal axes were found using the methods described in the supplementary material for Ref. . The sample exhibits a single phase transition at 7 K.
Experimental Techniques. THz spectroscopy measurements were performed at Lawrence Berkeley National Lab in a 7 T Janis Instruments magneto-optical cryostat. Further details on THz generation and detection are described in the supplementary material, section I. X-ray microdiffraction Tamura et al. (2003) on a 6 x 5 mm area of the sample was performed at the Advanced Light Source beam line 12.3.2. A representative Laue pattern from this region is shown in the supplementary material, section II. shown in Fig. 3 was measured by low-frequency susceptometry at ORNL.
Linear Spin Wave Theory. LSWT results were carried out at Berkeley using custom built software. Further details are found in the supplementary material, sections VI - X.
Acknowledgements: We thank N. Tamura and C. V. Stan for support at the Advanced Light Source beam line 12.3.2 and E. Angelino for help processing the x-ray diffraction data. We thank T. Scaffidi for useful discussions. Terahertz spectroscopy was performed at Lawrence Berkeley National Laboratory under the Spin Physics program (KC2206) supported by the US DOE, Office of Science, Office of Basic Energy science, Materials Sciences and Engineering Division under Contract No. DE-AC02-05-CH11231. Device fabrication and dc conductivity measurement were done at Stanford University under the Spin Physics program supported by the US DOE, Office of Science, Office of Basic Energy science, Materials Sciences and Engineering Division under Contract No. DE-AC02-76SF00515. A.L. and L.W. were supported by the Gordon and Betty Moore Foundation’s EPiQs Initiative through the Grant No. GBMF4537 to J.O. at U.C. Berkeley. The work at ORNL was supported by the US DOE, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division (J.Q.Y. and C.B.), and Division of Scientific User Facilities (A.B. and S.E.N.) under contract number DE-AC05-00OR22725. P. L. K. and D. M. acknowledge support from Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4416. E.A. acknowledges support from the ERC synergy grant UQUAM. D.B.’s participation in this research was facilitated in part by a National Physical Science Consortium Fellowship and by stipend support from the National Institute of Standards and Technology.
Author Contributions: L.W. conceived of the experiment. E.E.A. and E.A. devised the theoretical model. J.O supervised the project. L.W., D.R., and A.L. performed the THz experiments. L.W., A.L., and D.R. analyzed the THz spectra. E.E.A. performed the linear spin wave theory calculations. C.A.B. and A.B. purified the commercial -RuCl powder, P.L.K. and J.Q.Y. grew the single crystals. P.L.K. performed the ac susceptibility measurements. E.T. and A.L. performed the x-ray experiments and E.T. analyzed the diffraction data. D.B. measured the DC resistivity. A.L., E.E.A., E.A., L.W. and J.O. wrote the paper. All authors contributed to discussions and commented on the manuscript.
L.W., A.L. and E.E.A contributed equally to this work.
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Supplementary Material for “Field evolution of magnons in -RuCl by high-resolution polarized terahertz spectroscopy”
Supplementary Information for “Field evolution of magnons in -RuCl by high-resolution polarized terahertz spectroscopy”
Appendix A THz generation and polarization
THz pulses were generated focusing an 780 nm ultrafast laser pulse onto an Auston switch, consisting of a dipolar electrode antenna patterned onto a semiconductor. An AC bias voltage is applied across the electrodes while the laser pulse excites free carriers in the semiconductor. The carriers are accelerated by the bias voltage, emitting THz radiation. The THz pulses are focused onto the sample by off-axis parabolic mirrors and the transmitted radiation is collected by a receiver antenna. The THz focus spot size is large, 5 mm, and as such only samples with large lateral dimension are suitable. To select the direction of the THz magnetic field, B, the antenna polarization is fixed parallel to either the or axes of minimum THz transmission, as described in the main text. A grid-patterned polyethylene polarizer parallel to the antenna is placed directly before the sample (but outside of the cryostat). The DC magnetic field is applied by a 7 Tesla split-coil superconducting magnet in the plane of the sample.
A description of the data analysis for time-domain THz spectroscopy and a discussion of how the THz absorption is related to imaginary part of the magnetic susceptibility may be found in the supplementary information for our previous paper Little et al. (2017). The same methods are used in this work.
Appendix B Scanning X-ray Micro-diffraction
In order to confirm the multi-domain character of our sample, we performed scanning X-ray micro-diffraction Tamura et al. (2003) on the same sample studied by THz. Using the Advanced Light Source beamline 12.3.2, a full Laue (i.e., polychromatic) diffraction pattern is collected at spots of 2 m diameter in a 6 mm 5 mm region of the sample. We index using the lattice parameters in the C2/m space group given in Johnson et al. (2015). A representative Laue pattern from a single point is shown in Fig. S3. The diffraction peaks corresponding to three monoclinic structural domains, oriented at 120 degree intervals with respect to the sample normal, are indicated in the figure. We find that the sample is multi-domain across the region of study.
Appendix C Measurement of DC resistivity
We measure the dc resistivity of a thin flake from the same growth as the sample measured in the main text down to T=100 K. This measurement was done at Stanford University by exfoliating flakes a few mm in size and of about 100 m in thickness. Samples were contacted using EpoTek H20E epoxy and gold wirebonding wire. The samples were mounted on insulating sapphire substrate in a ceramic chip carrier (Kyocera PB-44567).Two-terminal DC transport measurements at low temperature showed insulating behavior, consistent with as shown in Fig. S2. We extend the range of this measurement to temperatures lower than previously measured, where at 100 K, cm.
Appendix D Temperature Dependence
To clearly plot magnon contribution from the full THz absorption, as shown in the main text Figs. 1 and 2, we subtract the spectrum at T = 8 K, just above the Néel temperature, from the spectrum in the ordered phase at T = 4 K. This removes absorption that is temperature independent across the magnetic transition, notably the large conductivity continuumLittle et al. (2017); Reschke et al. (2017); Bolens et al. (2017); Bolens (2018). Figure S3 shows the differential spectra for just magnon in the parallel configuration. The 8 K - 10 K spectrum shows the change in absorption is nearly zero, and thus we select the 8 K spectrum as a reference.
We further note that for the sample studied, the temperature dependance of the conductivity continuum does not show a clear trend and is very weak. In Fig. S3, we show differential spectra, with the T = 60 K spectrum as a reference for the up to 150 K.
Some samples have been shown to exhibit a structural phase transition in the range of 140 -160 K between monoclinic and trigonal or rhombohedral structures at low temperature Glamazda et al. (2017), while some have been shown to be monoclinic at low temperature Ziatdinov et al. (2016). Such a transition is sensitive to the stacking of the layers, and may appear hysteretic in temperature in the far infrared transmission/absporption Wang et al. (2017a); Reschke et al. (2017). The THz spectra of the sample used in this study do not show clear hysteresis related to the structural phase transition in this temperature range. We note that the temperature dependence of the broadband component of the absorption measured on this sample differs from that reported in Ref. Wang et al. (2017a).
Appendix E Determination of Magnetic Susceptibility
The total volume magnetic susceptibility, was measured by low-frequency susceptometry at Oak Ridge National Laboratory and was reported CGS units of emu/cm for each value of magnetic field. The measured by time-domain THz spectroscopy in SI units, where it is naturally a dimensionless quantity. To convert for direct comparison, we use the molar volume of -RuCl, 54.6 mol/cm and a factor of 4 between the two systems of electromagnetic units.
To extract the magnon contribution to from the THz spectra, we employ a the procedure described in the supplementary material for Little et al. (2017). In the parallel channels only absorbs, and thus it is straightforward to subtract the raw absorption spectrum at T = 4 K and 0 T from the spectra at H = 1 - 7 T – removing any field-independent features. We fit the residual spectra to the subtraction of two Lorenztian lineshapes as detailed in Little et al. (2017). This also allows us to remove unwanted systematic features of the raw data, such as oscillations in the frequency domain which result from time-domain pulse reflections.
In the perpendicular orientations, the appearance of the additional peaks and and make the field subtraction process described above unfeasible. In this case, instead was determined by fitting a Lorentzian function to the the differential spectra across the Néel temperature, for example as in Fig. S3. The fits are performed at each magnetic field step, and peak parameters for each of the four absorption features are extracted.
Using these fit parameters, we evaluate the sum rule integral relating to (Eq. (2) of the main text) for each absorption peak. The values for in main text Fig. 3 represent the total for all peaks, including both single-magnon and two-magnon contributions.
Appendix F Linear Spin wave theory calculations
As discussed in the main text, we employ linear spin wave theory to model the antiferromagnetic resonance modes observed by THz spectroscopy. Here we discuss a few further details of this calculation. Beginning from the Hamiltonian in Eq. (3) of the main text, our spin wave approximation represents the dilute limit of magnon fluctuations above a classical spin configuration. Such an approximation is accomplished by rewriting the local longitudinal spin component as polarized minus a number operator. Consistency of the commutation relations of the spin variables and new bosonic variables fixes the rest of the dictionary:
Where is the spin magnitude and the approximation in Eq. (2) and (3) is assumed in LSWT. These redefinitions are necessarily local when the magnetic order is not ferromagnetic. This technique is detailed in the more general case of incommensurate structures by Toth et al. Toth and Lake (2015). The classical spin configuration is obtained by assuming a given zigzag order and minimizing the presented Hamiltonian. We then expand in our bosonic operators around this classical configuration. The accuracy of this approximation requires that the spin deviation remains small on the spin size.
Making these substitutions yields a bosonic theory with interactions, which we neglect, resulting in a quadratic theory. The quadratic Hamiltonian is not number conserving and can be solved with a Bogoliubov transformation using the technique of Colpa et al.Colpa (1978) The zero point quantum fluctuations of the spins are captured in this approach and lead to reduction of the magnetic moment compared to its classical value. This reduction of static moment is the correction that we include when calculating the theoretical DC magnetic susceptibility (see the main text Fig. 5) and it is important in not overestimating this quantity.
The dynamic structure factor is given by the two spin correlation function which can be calculated with free field correlators in the bosonic language. This is related to absorption with linear response theory.
In the bosonic language this expression amounts to the evaluation of two and four point functions in a free theory. The transverse contributions are exclusively two point functions and should therefore only be sensitive to, in this approximation, the one magnon spectrum. The longitudinal component of each spin contains a number operator so evaluating the longitudinal absorption involves a four point function. A four point function in free field theory splits into an integral over pairs (in our case with net momentum zero) and necessarily generates a continuum response. It is interesting that this continuum can generate contributions that look sharp enough in width to be confused with higher energy spin wave modes or other bound states.
For our Hamiltonian, the 1/S corrections to spin-wave theory are small but not insignificant. They are strongest at 0-field, and close to the 7 T transition. The 0-field corrections come from spin flip occupation at momentum given by the wave vectors of the unchosen zigzag orders. The zero field corrections are due to soft fluctuation modes at the right wave-vector that would take one zigzag configuration to one of the other two degenerate states. These corrections however should not couple strongly to the individual = 0 modes.
Near the transition, the zero point fluctuation (namely, the occupation of the Holstein-Primakoff bosons ) is almost exclusively at = 0 and we expect quantum corrections to our calculated spectra.
Appendix G Q-Flop
As a simple model of the anisotropy present in -RuCl we consider an average of LSWT results on three honeycomb patches. On each patch a different bond type is “lengthened” by reducing all interactions across this bond by 3. This results a 0-field selection of a given zigzag order (with wave vector parallel to the stretched bond), and a competition between the magnetic field and the anisotropy of the lattice. We implement this competition by assuming the zigzag order with the minimum classical energy in each patch. This model demonstrates the physical mechanism for the abrupt change in absorption near 1.5 T as a Q-flop, a shifting of zigzag order. The average over patches is necessary for this jump to be observed for any direction of applied field. The scale of this crossover, and the fact that it occurs for multiple directions of in-plane applied field sets constraints for future studies which will necessarily consider bond-dependent couplings. Such a refined model must not select a given zigzag order too strongly, otherwise an applied field along the preferred zigzag order would yield no low field discontinuous behavior.
Appendix H Symmetries and Selection Rules
We find the Hamiltonian enjoys two residual symmetries even in presence of the zigzag order.. The first is a sublattice symmetry. In a zigzag order the sublattice contains four spins, two on each ferromagnetic strip. The sublattice symmetry is a simultaneous switching of spins within each zigzag strip. This symmetry acts simply on the = 0 modes and they transform under representations of it. The higher energy modes are odd under this transformation and the lower energy modes are even. A uniform magnetic field is even under this transformation so it can only couple to the lower modes. This explains the absence of a response from the two higher spin wave modes.
The second selection rule is the result of a spin-space symmetry of the zigzag state. This symmetry exchanges the two spin directions not associated with the bond that joins ferromagnetic strips. For instance, if the bond that joins the ferromagnetic strips is z-type, then at zero external field there is a symmetry upon the exchange of the x and y spin coordinates. This symmetry is approximate at finite field and exact for zero field. One of the lower modes, is odd under this transformation, while is even. The in-plane probe field, B, if applied parallel or perpendicular to the ordering wave vector will be respectively odd or even under this transformation and couple to a mode of the same parity.
Appendix I LSWT spectra for
The theoretical absorption for is shown in Fig. 2 of the main text. Here we present the orientation, in the parallel and perpendicular channels for a multi-domain model. An important distinction between these cases is that a field along the axis selects = Y (z-bond stretched) and is parallel to it so the selection rules apply, whereas a field along the axis selects = M,M’ (x or y bond stretched) and is neither parallel or perpendicular to these directions. We thus see some mixing of the polarization of and ; both modes are visible even after the 1.5 T crossover. Another way to visualize this is to consider the fact that zigzag orders can be rotated into each other with a joint spatial and spin rotation of 60 degrees and 120 degrees about the out of plane direction respectively. This action connects the zigzag strips which are angled 60 degrees relative to each other and appropriately changes the bond types. Therefore the spin wave modes which exclusively couple to directions or will be rotated 120 degrees into modes which have mixed absorption with respect to these directions. This effect weakly observed in the experimental data shown in the main text Fig. 2 (b), where persists up to 3 T. As mentioned in the main text, the experimental results indicate that the modes broaden more rapidly for the direction than for the . Classically, our model has a transition for fields along the direction at around 7.5 T and for the direction at 10 T. However the exact diagonalization results of Winter et al. Winter et al. (2018) suggest the difference in critical field between the two directions will be smaller after accounting for quantum effects. For this reason we expect that for fields of about 7T, in the direction of , the, corrections beyond semiclassical linear results become relevant.
Appendix J Momentum dependence
We note that the minimal LSWT model we use to describe the THz spectra at = 0 does not reproduce a feature present in the INS data at higher values of Q. The low field spectra in Banerjee et al. (2018) show that long the line in momentum space from the point to the M point, the lowest spin wave mode appears to increase in energy before decreasing again. This feature is not present in the standard well-studied models. The dispersion of the modes across the Brillioun zone for our LSWT model, at = 2 T, is shown in Fig. S5, where local minima appear at the M-points but not the -point.
We find that a local minimum at the -point may be introduced by adding a second nearest neighbor ferromagnetic Heisenberg term on bonds perpendicular to the order wave vector on the order of .3 mev. Anisotropic second neighbor terms of this order and larger are predicted by ab initio studies Winter et al. (2016). Including such terms has the additional benefit of increasing the M-point gap closer to observed values. We would like to emphasize that in modeling the full dispersion, such terms shouldn’t be ignored due to their small magnitude because for each site, there are six second neighbor couplings. Further study is required to fit such terms to experimental data.