APS/123-QED
Local Magnetism and Spin Dynamics of the Frustrated Honeycomb Rhodate Li2 RhO3 P. Khuntia∗ , †,1 S. Manni‡,2 F. R. Foronda,3 T. Lancaster,4 S. J. Blundell,3 P. Gegenwart,2 and M. Baenitz1 1 Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany 2 EP VI, Center for Electronic Correlations and Magnetism, Augsburg University, D-86159 Augsburg, Germany 3 Oxford University Department of Physics, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom 4 Durham University, Centre for Materials Physics, South Road, Durham, DH1 3LE, United Kingdom (Dated: updated version on:; Received textdate; Revised textdate; Accepted textdate; Published textdate)
arXiv:1512.04904v2 [cond-mat.str-el] 16 Dec 2015
Abstract We report magnetization, heat capacity, NMR, and µSR studies on the honeycomb rhodate Li2 RhO3 , which is a candidate material for Kitaev magnetic exchange. The magnetic susceptibility displays Curie Weiss behavior with θCW = – 60 K and an irreversible behavior below 6 K in small magnetic fields. The heat capacity measured in various applied magnetic fields exhibits no sign of long range ordering down to 0.35 K but a broad maximum at about 10 K. Magnetic heat capacity (Cm ) is field independent and follows a T 2 behavior below 10 K. The7 Li NMR line width and the spin-lattice relaxation rate (1/T 1 ) exhibit signatures of short range spin correlations and slowing down of spin fluctuations. The power law behavior of 1/T 1 (∼T 2.2 ) and C m (∼ T 2 ) below 10 K suggests the presence of low lying gapless excitations. µSR results in zero field also reveal the persistence of quasi-static moments and short range spin correlations. Our comprehensive results demonstrate a quantum disordered and non-singlet ground state in this honeycomb lattice. PACS numbers: 75.10.Jm, 75.10.Kt, 76.60.-k, 76.60.Es, 76.75.+i
Strongly correlated electron systems have provided a plethora of novel phenomena stemming from competing interactions driven by charge, spin and orbital degrees of freedom [1–3]. Highly frustrated S = 1/2 quantum magnets (QM) host unconventional magnetic properties and they pose an enduring theme of the modern condensed matter physics. The physics of S = 1/2 QM is rich and relies mainly on the spin degrees of freedom and the magnetic exchange interactions determined by the lattice geometry and the orbital hybridization. Prominent candidates are quasi 1D- linear chains, planar 2D-models (ladders, triangles or square lattices) or more complex structures like the kagome- or pyrochlore lattices [1–4]. Recently, the field of S = 1/2 quantum magnets have been extended towards higher d- states. Apart from 3d states residing on Cu2+ or V4+ ions, an effective S = 1/2 moment could be realized in some 4d- and 5d- systems. The spin-orbit (SO) interaction, which is a relativistic quantum effect, gained significant attention following the discovery of topological insulators wherein SO interaction induces non-trivial topological quantum states [5, 6]. Here the sizable SO interaction in combination with the Coulomb interaction and the crystal electric field splitting lead in some cases to a “ pseudo” S = 1/2 state. Beside geometrical frustration, strong exchange frustration is frequently found in these systems and a novel ground state, quantum spin liquid (QSL) is postulated [1, 2]. The excitations in the QSL state can be gapless or gapped with topological order and fractional excitations, but a general consensus concerning the exact nature of these excitations is still a debatable issue because of disorder, anisotropic interactions, and scarcity of model materials [1, 7–10]. A prominent example in this context is the Heisenberg - Kitaev model, which found its experi-
mental realization in some honeycomb lattice systems like Li2 RhO3 , (Li,Na)2 IrO3 , α− RuCl3 and proposed to host exotic excitations pertinent for the search of exotic states such as QSL and to test theoretical models [1, 4, 9, 11–27]. In analogy to linear dispersive (∼ E ) electronic states in graphene in the Dirac QSL, the magnetic excitations are driven by linear dispersive low lying states and this gapless state is one of the hallmark of a magnetic QSL [28, 29]. The 5d iridates and particularly the honeycomb lattice systems (Na/Li)2 IrO3 display intriguing magnetic properties which are driven by strong SO interaction with comparable spin correlations [11, 13, 14]. Na2 IrO3 displays zig-zag magnetic ordering, while Li2 IrO3 exhibits incommensurate spiral ordering [30–32]. Further more, a Kitaev exchange between effective jeff = 1/2 moments is discussed in α− RuCl3 and the planar honeycomb material α−Li2 IrO3 , and also in other polymorphs [33, 34]. The magnetic exchanges in Na- and Li-system could be originated from geometrical frustration of the Heisenberg interaction and dynamical frustration of Kiatev interaction [35, 36]. Surprisingly, their structural 4d-homologue Li2 RhO3 also shows insulating behavior in spite of reduced SO and even more interestingly this system shows no LRO unlike its Ir counterpart. Rather it shows a spin-glass like freezing at low temperature, which could be driven by magnetic frustration and negligible but inevitable stacking faults [31]. Magnetic exchange between Rh4+ ions are expected to be highly frustrated and the observation of effective jeff = 1/2 moments renders the possibility of Kitaev exchange in this material [1, 31, 37, 39]. In view of the presence of competing magnetic interactions and SO interaction, Li2 RhO3 is proposed to be a promising quantum 1
magnet to display novel ground state properties. In this Letter, we provide a comprehensive account of the local magnetic properties of Li2 RhO3 probed by 7 Li (I = 3/2) NMR and muon spin relaxation (µSR) accomapnied by magnetization and specific heat studies. Li2 RhO3 displays no signature of long range magnetic ordering down to 0.35 K but a short range spin-glass like behavior. The magnetic specific heat and spin lattice relaxation rate exhibit power law behavior at low T implying gapless excitations. Our investigation reveals a quantum disordered and non-singlet ground state with quasi-static moments down to 0.35 K. Shown in Fig. 1(a) is the temperature dependence of the dc magnetic susceptibility χ(T ) measured following ZFC and FC modes at 100 Oe. The ZFC-FC χ(T ) curves split at 6 K, which is indicative of a spin freezing. However, this splitting is small, which indicates that a fraction of spins participate in the glassy state and a similar scenario has been discussed in a variety of quantum magnets [18, 21, 22, 25–27]. The Curie-Weiss (CW) fit (Fig. 1a inset) in the temperature range 200 ≤T ≤ 300 K yields an effective moment 2.2 µB which is slightly larger than Rh4+ spin (S = 1/2) only value, which suggests the presence of moderate spin-orbit interactions in the system [1, 37, 39]. The negative value of θCW (≈ – 60 K) determined from the CW fit suggests the presence of AFM correlations between Rh4+ spins. The effective exchange coupling between Rh4+ using mean field theory can be estimated as: J /kB = −3θCW /zS(S + 1) = 80 K (with coordination number, z = 3), which is a very simple and crude estimate given the presence of various exchange couplings in Li2 RhO3 of comparable strengths [1, 30]. The high temperature series expansion (HTSE) valid for honeycomb lattice also yields an antiferromagnetic interaction of J /kB ≈ (75 ± 5) K between Rh4+ spins [40]. The ac susceptibility exhibit a peak at about T g ≈ 6 K, (see Supplemental Material [41]) and the peak positions are frequency dependent (albeit weak), inferring the role of dissipative spin dynamics in driving such a short range spin freezing mechanism. The origin of this fractional spin freezing could be ascribed to the presence of minority spins, frustration in addition to defects/and or stacking faults in the Rh-honeycomb stacking in the host lattice [13, 14, 31, 39]. A small fraction of orphan spins commonly found in frustrated materials Na4 Ir3 O8 , Ni2 Ga2 S4 , SrCr9p Ga12−9p O19 , and Cu3 BaV2 O8 (OH)2 [17, 18, 20–22, 26, 27, 42]. The glassy feature in Li2 RhO3 smears out with an applied field of 1 T and χ remains finite at low T (see Fig. 1(a) inset), which suggests a slowing down of spin dynamics and a field polarized state. The finite value of χ at low T is indicative of a non-singlet ground state and reflects that the majority of spins are correlated and drive the low T magnetic properties discussed henceforth. The heat capacity measured in different magnetic fields is shown in Fig. 1(b). The heat capacity exhibits no signature of LRO down to 0.35 K. The magnetic heat capac-
ities (Cm ) in different magnetic fields were obtained by subtracting the lattice contribution using Li2 SnO3 (see Fig. 1(b)) as a reference. As shown in Fig. 1(b), Cm /T displays a broad maximum at about 10 K, which could be associated with highly frustrated nature of the system as discussed in spin liquids [1, 26, 43]. The strength of the exchange coupling and dimensionality of the system accounts for the position of the broad maximum in specific heat and it varies as T/θCW in frustrated magnets. In Li2 RhO3 , we found T/θCW ≈ 0.16, which is comparable with those found in other 3d and 5d frustrated magnets [17, 18, 26]. This suggests the co-existence of frozen and slowly fluctuating spin dynamics induced by competing magneticR interactions in this rhodate. The magnetic entropy Sm = Cm /T dT thus obtained (≈ 2.04 J/mol-K2 as shown in Fig. 1(b) inset) is about 35% Rln2 (5.76 J/mol-K2 ) at 45 K. The loss of magnetic entropy compared to the expected value for S = 1/2 is due to short-range spin correlations and suggests that the low lying states are highly degenerate. The magnetic heat capacity (Cm ) exhibits a broad maximum at 10 K, below which shows a T 2 behavior indicating the persistence of spin dynamics with low lying gapless excitations, which is in agreement with the finite value of χ at low T. The T 2 dependence of Cm is typical for short range ordering phenomena as in the case of a few 3d and 5d frustrated magnets appears due to collective excitations yielding novel ground states [17, 18, 21, 25–27]. It is worth mentioning here that the T 2 dependence of magnetic heat capacity is attributed to a spin liquid state with spinon Fermi surface for the 5d iridate Na4 Ir3 O8 [44, 45]. The 7 Li NMR powder spectra at 70 MHz shown in Fig. 2(a) clearly depicts broadening towards low temperature due to a distribution of hyperfine fields. The inset of Fig.2(a) represents the T-dependent NMR shift, K (T) estimated from the simulation of each spectrum. We found a very weak T-dependence down to 1.8 K and the origin of this behabvior of K could be ascribed to spin orbit interactions [45–47]. The linear scaling between K and χ (K=Ahf χ/N A ) at high T yields a hyperfine coupling constant Ahf = – 0.3±0.06 kOe/µB between 7 Li nucleus and the Rh4+ electron spin. The finite value of K at low T suggests a non-singlet state in agreement with the χ results. Shown in Fig. 2(b) is the NMR line width (FWHM) normalized to reference fields (relative line width, δH = ∆H/H) at two frequencies. The relative line width, δH is enhanced in the intermediate temperature regime without signature of LRO and δH is independent of T at low temperature, which is in agreement with our χ results. The low T broadening of the NMR spectra is associated with the inhomogeneous nature of local magnetic fields at the nuclear sites and slowly fluctuating hyperfine fields. It is remarkable that δH is independent of magnetic field. This implies that the absolute shift is field dependent, which suggests that at these fields the system is not yet in the fully polarized state and correlations among Rh4+ spins are still present. This is consistent with the absence of LRO in Cm 2
the Rh4+ spins at the 7 Li sites, and taking K as the intrinsic susceptibility in place of χ. One would expects T11T χ ≈ 1/ωc when ω c ≫ ω N , and for ω c ≪ ω N , T11T χ should be magnetic field dependent. When ω c = ωN , both T1 T1 K and 1 approach maxima (Fig.3(b)), which appears as a conT1 T χ sequence of the slowing down of the fluctuation frequency ω c of Rh4+ spins. We find that ω c is nearly T -independent at high temperature, but decreases below 10 K with ωc ∝ T 1.2 behavior at low-T suggesting the slow spin dynamics of Rh4+ , which is consistent with the broad NMR line at low temperature. The slowing down of spin flctuations might dictate the low temperature magnetic properties [21, 22]. The low T behavior of 1/T 1 suggests damping of low energy excitations without signature of static magnetic ordering. The power law dependence of 1/T 1 in Li2 RhO3 could be compared with those found in Na4 Ir3 O8 [46] and other quantum magnets [7, 24, 57–59], which is attributed to the existence of gapless state in the spin excitation spectrum and in accord with finite value of χ and K , and Cm ∼T 2 behavior at low T. One would expect T-independent behavior of 1/T 1 T at low T in the case of a fully gapless excitation with spinon Fermi surface taking into account the observed Cm ∼T 2 behavior [7, 44–47]. It may be noted that the power law behavior of dynamic spin susceptibility (1/T 1 ≃ T n ) is one of the challenging and debatable issues in the context of field theoretical approach in interpreting quantum materials in fermionic model. The gapless excitations in Li2 RhO3 may correspond to an instability of the spinon Fermi surface [44, 45, 60, 61]. Our muon-spin rotation (µSR) [62] experiments on a polycrystalline sample of Li2 RhO3 were carried out on the general-purpose spectrometer (GPS) at the Swiss Muon Source (SµS) at PSI (Switzerland). Representative spectra are shown in Fig. 4(a). For temperatures below about 6 K, there is a single heavily damped oscillatory signal (proportional to cos(2πν µ t)e−Λt ) together with a slow relaxation (proportional to e−λt ). The damped oscillation signifies static magnetic Rh4+ spins but the large damping is entirely consistent with spin freezing, and is not associated with long range magnetic order. The frequency of the damped oscillation, ν µ , is around 2 MHz and falls slightly on warming [see Fig. 4(b)] while the relaxation rate λ rises. At temperatures above ≈ 6 K, it is no longer possible to fit the fast relaxation with a damped oscillation, and we identify this temperature with the spin freezing temperature Tg , in agreement with magnetization measurements [1, 37] and coinciding with the peak measured in ac susceptibility (see Supplemental [41]) . For T > Tg we fit our data instead using a sum of two exponential relaxations, so that the fitting function becomes A(t) = A(0)[Af e−Λt + (1 − Af )e−λt ]. The amplitude of the fast relaxing term Af falls on warming above 6 K [see Fig. 4(c)] and is entirely absent by 15 K by which temperature the relaxation is dominated by a Gaussian response
down to 0.35 K despite θ CW ≈ –60 K, which indicates the presence of a quantum disordered state in Li2 RhO3 . The saturation of δH at low T indicates the persistence of quasi-static inhomogeneous distribution of magnetic fields and slowing down of electron spin fluctuations (but not critical slowing down) wherein Rh4+ spins fluctuate with a frequency less than the NMR frequency. We estimate a magnetic moment of 0.07 µB , which is very small compared to the Rh4+ moment and suggests the presence of quantum fluctuations induced by magnetic frustration at low T [22, 24]. The NMR spectra do not exhibit any signature of quadrupolar effects. Furthermore, δH scales with χ (as shown in Fig.2(b) inset) much above the spin-glass transition suggesting a dominant role of dipolar interactions. The T dependence of NMR signal intensity provide information concerning the nature of hyperfine coupling and fluctuating moments in the intermediate and low T regimes and because of slowing down of spin fluctuations one would expect a loss in NMR signal intensity typical in cuprates, spin glass and low dimensional magnets [48–52]. In the present case, we found no loss of NMR signal intensity, which indicates that Li2 RhO3 is not a conventional spin glass material and suggests the absence of LRO. This scenario is further supported by the absence of rectangular shaped poweder averaged NMR spectra expected for LRO materials [53]. This supports the fact that a fraction of spins due to defects/or stacking faults responsible for glassy behavior observed in magnetization and smears out with magnetic field of a few Teslas and not a bulk effect, and a majority of spins remain fluctuating at low T [18, 21, 22, 25–27]. The spin lattice relaxation rate (1/T 1 ) probes the wave vector averaged low energy spin dynamics. Shown in Fig. 3(a) is 1/T 1 vs. T at two different frequencies. 1/T 1 decreases with T and passes through a broad maximum around 10 K suggesting the presence of short range correlations and reduction of spin excitations driven by frustration. In addition, a plateau in the T-range 30 ≤T ≤ 100 K and a broad maximum in 1/T 1 are signatures of slow spin dynamics which is in contrast to that of conventional spin glass magnets wherein critical slowing down of spin fluctuations lead to a very short T 1 and NMR signal wipeout at low and intermediate temperatures [22, 24]. As shown in Fig. 3(a), 1/T 1 decreases on further cooling below 10 K and displays a T 2.2 behavior down to 1.8 K. In principle, 1/T1 tracks the spectral density of the Fourier transform of the time correlation function of the transverse component δh± of the fluctuating local field at nuclear sites with the nuclear Larmor frequency ω N as [54– γ 2 R +∞ 56] T11 = 2N −∞ h h± (t)h± (0) ieiω N t dt , where γ N is the gyromagnetic ratio of the nuclear spin. Assuming the time correlation function varies as e−Γt , one can express Γ 1 = A ω2 +ω 2 (eq.1) where A is a parameter associT1 T χ c N ated with the hyperfine field at nuclear sites and χ is the magnetic susceptibility. Here, ω c corresponds to the correlation frequency of the fluctuating hyperfine fields due to 3
characteristic of static nuclear moments. These observations indicate that although the spin freezing disappears above Tg there remain some frozen regions of the sample which persist well above Tg , up to almost 2Tg , perhaps in small, slowly-fluctuating clusters. These slow fluctuations may likely contribute to the slow relaxation that is observed in these data. The volume fraction of the clusters decreases on warming and this can be directly related to the decrease in Af . In fact, the observation of a slowly relaxing fraction throughout the temperature range demonstrates that the spin frozen state possesses some weak dynamics. These measurements are consistent with the development of a spin-frozen state below 6 K at small magnetic fields. To summarize, the magnetic susceptibility (χ) exhibits AFM interaction between Rh4+ moments with J /kB ≈ (75 ± 5) K. In Li2 RhO3 , χ, δH and K remain finite at low T suggesting a non-singlet behavior. The magnetic heat capacity (Cm ) displays no signature of long range magnetic ordering down to 0.35 K and Cm ∼T 2 implies the gapless nature of low lying states. 1/T 1 ∼ T 2.2 behavior at low-T infers slowing down of spin dynamics and gapless excitations consistent with magnetic heat capacity. The current investigation sheds light on the role of quasi-static moments and correlated spin dynamics of majority spins in driving exotic quantum disordered state in Li2 RhO3 . The present work will stimulate further explorations in SO driven quantum magnets. We acknowldge insightful discussions with A. V. Mahajan, H. Yasuoka, and M. Majumder. PK acknowledges support from the European Commission through Marie Curie International Incoming Fellowship (PIIF-GA2013-627322). SM acknowledges financial support by the Helmholtz Virtual Institute 521 (“New states of matter and their excitations”). FRF, TL and SJB acknowledge support from EPSRC (UK). Part of this work was carried out at the Swiss Muon Source (SµS), Paul Scherrer Institut, Villigen, Switzerland. †Present address: Laboratoire de Physique des Solides, Universit´e Paris-Sud 11, UMR CNRS 8502, 91405 Orsay, France. ‡Present address: Department of Physics and Astronomy, Iowa State University, Ames, Iowa, USA.
Li2 RhO3 at different temperatures and the solid line is the simulation. The inset shows the T dependent NMR shifts (-K) at 70 MHz and 114 MHz (b) The T dependence of full width at half maximum (FWHM) normalized to reference fields (relative line width, δ H=∆H/H) at 70 MHz and 114 MHz compared with that of its non-magnetic analog Li2 SnO3 . The inset shows δ H vs. χ with temperature as an implicit parameter. Fig. 3 (Color online) (a) The T dependence of 1/T 1 at two frequencies and the solid line is the T 2.2 behavior (b) T dependence of 1/T 1 T χ and the solid line represents the T 1.2 behavior. (c) The recoveries of longitudinal magnetization M(t) with stretched exponential fit at various temperatures. Fig. 4 (Color online) (a) Representative µSR spectra for Li2 RhO3 measured at a range of temperatures. (b) The fitted precession frequency ν µ which indicates the average field experienced by the muon in the spin frozen state (solid black circles, left-hand axis) and the relaxation rate λ of the slowly-relaxing component (open red squares, righthand axis); (c) the amplitude Af of the fast-relaxing component (solid black circles, left-hand axis) and λ (open red squares, right-hand axis).
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*Email:
[email protected] Figure Captions: Fig. 1 (Color online) (a) The temperature dependence of ZFC-FC magnetic susceptibility (b) The Honeycomb lattice of Li2 RhO3 . (c) χ vs. T at 1 T with HTSE and CurieWeiss fit as discussed in the text (d) The temperature dependence of heat capacity in zero field for Li2 RhO3 compared with its non- magnetic analog Li2 SnO3 . (e) T/ θ CW dependence of magnetic heat capacity co-efficient (Cm /T) in different applied magnetic fields and the solid line is a fit as discussed in the text. (f) T dependence of magnetic entropy in zero field. Fig. 2 (Color online) (a) Field swept 7 Li NMR spectra in 4
is isoelectronic and isostructural with (Na/Li)2 IrO3. Edge sharing RhO6 octahedra forms Rh-honeycomb lattice in the Li2 Rh plane where Rh carries J ef f = 1/2 moment in Li2 RhO3 (shown in Fig.1(a) inset). The spin polarized calculations show that the ground state is degenerate and stable between FM and AFM phases [1]. The dc magnetic susceptibility (χ=M/H) were measured using a SQUID magnetometer (MPMS, Quantum Design, QD) in thetemperature range 1.8 ≤ T ≤ 300 K in different magnetic fields. The ac susceptibility data taken employing an ac susceptometer (PPMS,QD) at different frequencies. It may be noted that this ZFC-FC splitting could be related to the role of a fraction of magnetic moments exhibiting glassy behavior and a similar scenario has been discussed in NiGa2 S4 ,Cu3 BaV2 O8 (OH)2 ,Ba3 CuSb2 O9 ,Na4 Ir3 O8 and Ba3 IrTi2 O9 . The hysteretical magnetization could be related to minority spins, which appear due to defects and/or stacking faults in the host lattice. The exchange interaction between the nearest neighbor Rh4+ spins in the first P approximation is described by the Hamiltonian, H = J Si .Sj . In order to estimate the exchange coupling (J ) between the nearest neighbor Rh4+ spins in the triangular lattice, we have the magnetic susceptibility data following the high temperature series expansion (HTSE) for S = 1/2 system appropriate for honeycomb lattice antiferromagnet 2 2 P n A g µB χ = N4k ( kBJT ) BT where n = 0..5 and an are series coefficients and the values of which can be found in [2] and we found J /kB ≈ (75 ± 5) K, which is close to that estimated using MFT. Given the simplicity of the model and appreciable spin orbit interactions with Jeff = 1/2, the obtained coupling constant is a rough estimate. It may be noted that the HTSE is valid for S = 1/2 systems without spin orbit interactions. The temperature dependence of heat capacity measurements in various magnetic fields and temperature down to 0.35 K have been performed in the 3 He option of a PPMS using the thermal relaxation method. NMR being a powerful local probe is useful to gain microscopic details pertinent to the local magnetism and dynamic correlations in Li2 RhO3 . The NMR spectra and spin-lattice relaxation measurements at two frequencies were carried out using a standard pulsed NMR spectrometer. The low value of magnetic moment at low T is in line with the strong magnetic frustration due to various magnetic configurations with comparable energy scales in Li2 RhO3 . The spin lattice relaxation measurements were performed following the saturation recovery method using short rf pulses. In Li2 RhO3 , the recovery of longitudinal nuclear magnetization, M(t) at time t after the saturation pulse could be fitted with 1β M(t)/M(∞) = e−(t/T1 ) (M(∞) is the saturation magnetization, and β (≈ 0.5) is the stretch exponent, which is found to be independent of temperature here), which is related to
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Supplemental Material: Local Magnetism and Spin Dynamics of the Frustrated Honeycomb Rhodate Li2 RhO3 Experimetal Details Polycrystalline samples of Li2 RhO3 were synthesized by a method described elsewhere [1]. The titled compound 5
of δH, 1/T 1 T K, and 1/T 1 T χ in the intermediate T regime is consistent with the presence of AFM spin correlations with q 6= 0 components of spin fluctuations. Whereas the T independent behavior of 1/T 1 T K and 1/T 1 T χ at T >80 K could be interpreted in terms of local moment magnetism with dominant q = 0 components of spin fluctuations. The exchange coupling rather than thermal fluctuation plays a key role in driving the spin dynamics of Li2 RhO3 at high temperature.
a distribution of T 1 values. The spin-lattice relaxation rate, 1/T1 at each temperature is determined by fitting the nuclear magnetization M (t) using the stretched exponential function. The non-exponential auto correlation functions is frequently found in spin liquids and but also in disordered materials. In general, 1/T 1 probes the q-averaged low energy spin excitations and 1/T 1 T is expressed as the wave-vector q summation of the imaginary part of theP dynamic electron 2 spin susceptibility χ′′ (q,ωn ): T11T ∝ q | Ahf (q) |
Fig. SM1 (Color online) The temperature dependence of the real part of ac magnetic susceptibility at various frequencies.
χ′′ (q,ω n ) ωn
, Ahf (q) is the form factor of the hyperfine interactions and ω n is the nuclear Larmor frequency [3]. The T dependence of δH and 1/T 1 in Li2 RhO3 have been compared with those of its non magnetic reference Li2 SnO3 . For Li2 SnO3 , the 7 Li NMR spectra are quite narrow and 1/T 1 T exhibits an activated behavior frequently found in other non-magnetic oxides. The relaxation rate, 1/T 1 is three orders of magnitude smaller than that of Li2 RhO3 . Shown in Fig. S2 is the temperature dependence of 1/T 1 T K . of Li2 RhO3 . It is observed that, both 1/T 1 T K and 1/T 1 T χ shown in Fig.3 (b) (χ is the uniform spin susceptibility) are independent of temperature at high T, while they are strongly enhanced in the intermediate T regime and exhibits T 1.2 behavior below 10 K. The enhancement
Fig. SM2 (Color online) The temperature dependence of 1/T 1 TK at two NMR frequencies
[1] I. I. Mazin et al., Phys. Rev. B 88, 035115 (2013). [2] H.E. Stanley, Phys. Rev.158, 546 (1967); G. S. Rushbrooke and P. J. Wood, Proc. Phys. Soc. A 68, 1161 (1955). [3] T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism (Springer, Berlin, 1985).
6
100
8
p
T
100
(K)
100 0 T
(K)
3 T
9 T
T
= 100 Oe
(b)
~
FC 2 (f) (J/mol-K)
ZFC
8
S
m
1
7
0 0
5
10
T
15 (K)
20
25 0.01
T
/
T
15 30 45 (K)
0.1 CW
T (10
H
0
J/mol-K )
10
1
m
1
3
6 T
-3
3
2
= 10 kOe
T
3
C /
H
10
(10
2
Li SnO
2
4
Li RhO
0.01
C-W fit
10
9
10
-2
-3
(10
HTSE
cm /mol)
11
(e)
(d)
1
3
12
(c)
C (J/mol-K)
cm /mol)
(a)
1 Hz
2.4
100 Hz 1 kHz
' (10
-5
emu/mol)
10 Hz
2.2
2.0
5
T (K)
10
15
(b)
114 MHz
(%)
= 70 MHz 0.84
K
250 K
8
6 4 2
0.78
10 K
T
100
0 0
(K)
2
4
(10
-3
6 3
8
6
)
50 K
10
cm /mol)
-3
1
4.3 K 1.8 K
4
simulation
H/H (10
100 K
H =
Echo Intensity
200 K
)
70 MHz
Li NMR,
-3
7
H (10
(a)
70 MHz (Li RhO ) 2
3
114 MHz (Li RhO ) 2
3
70 MHz (Li SnO ) 2
4.1
4.2
H
0
4.3 (T)
4.41
10
T
3
100 (K)
2
1/
1
T TK (s
-1
K
-1
)
1
0.1
70 MHz 114 MHz
T
1.2
0.01 1
10
T (K)
100
1
T (K)
10
10
100
3
.mol/cm )
100
(b)
(a)
70 MHz 114 MHz
-1
K
-1
10
(s
70 MHz 114 MHz 1
1.2
1
TT
T
-1
)
1/
1
1-
0.1
( )/
Mt M
(0)
fit
(c) 1
120 K
25 K
10.1 K
2.2
5 K
T
~
2.5 K
1/
T
1.8 K
1
(s
0.1
0.1
0.01
1
T (K)
10
0
25
t (s)
50
75
100
15.0 K
(b)
9.4 K
νµ (MHz)
8.4 K
6.9 K
(c) 6.0 K
Af (%)
Asymmetry (%)
(a)
1.7 K
