Kai-Yue Zeng(曾凯悦), Long Ma(马龙), Long-Meng Xu(徐龙猛), Zhao-Ming Tian(田召明),‡,Lang-Sheng Ling(凌浪生), and Li Pi(皮雳),§

1Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions,High Magnetic Field Laboratory,Chinese Academy of Sciences,Hefei 230031,China

2Hefei National Laboratory for Physical Sciences at the Microscale,University of Science and Technology of China,Hefei 230026,China

3School of Physics and Wuhan National High Magnetic Field Center,Huazhong University of Science and Technology,Wuhan 430074,China

Keywords: low-dimensional quantum magnetism,magnetic coupling,spin excitations,nuclear magnetic resonance

1. Introduction

The strong quantum fluctuations in the low-dimensional(low-D) antiferromagnets always lead to exotic quantum excitations as well as attractive quantum ground states.[1–4]One of the archetype examples is the spin-1/2 Heisenberg antiferromagnetic(AFM)chain system,which is proved exactly to host the quantum critical ground state called Tomonaga–Luttinger liquid.[5]For the spin excitations of such systems, the gapless “domain wall” excitations with fractional quantum number called spinons are theoretically proposed and further experimentally verified.[6]

After the pioneering work of Haldane in 1983,[7,8]people began to realize that chains with integer-spins (Haldane chains) are topologically different from those with half-odd spins. The ground state of Haldane chains is quantum disordered singlets, which can be viewed as a simple version of the valence bond solid state. In contrast with the gapless excitation continuum of the multi-spinon excitations,the excitations from singlets to triplets need to overcome a finite energy gap(Haldane gap).[7,8]Interestingly,the spin excitation spectrum of Haldane chains is asymmetric about the Brillouin zone boundary as a result of the maintaining translational symmetry of the crystal lattice,[9–11]very different from the ordinary N´eel ordered state. Up to now, several quasi-1D S=1 quantum magnets have been identified as ideal realizations of the quantum singlet ground state.[12–18]

By further introducing the inter-chain coupling (J⊥) and single-ion anisotropy (D) to the integer spin Hamiltonian, a rich phase diagram can be reached in the J⊥–D space.[19,20]Novel ground states, such as Bose–Einstein condensation of magnons,[1,20,21]spin supersolids,[22,23]can be realized by tuning the competing interactions. By applying external magnetic field or physical pressure,quantum phase transitions can be triggered in such low-D spin systems. As a result, integer spin chain systems with complex structures and magnetic couplings have supplied valuable playgrounds for the condensed matter society to explore novel quantum excitations and criticality.

The AFM insulator Ni2NbBO6, first synthesized about four decades ago,[24]exhibits complex lattice structure and magnetic interactions. The lattice structure can be viewed as coupled armchair spin chains along the crystalline b-axis, or zig-zag spin chains along the c-axis,where the magnetic Ni2+sites carry integer S = 1 spins. DC-susceptibility measurements indicate an AFM transition at TN∼23.5 K for a low magnetic field and a field induced spin-flop transition nearµ0H ∼3.67 T in the low temperature ordered phase, when the field is applied perpendicular to the armchair spin chain direction.[25]With density functional theory (DFT) calculations,the magnetic coupling configuration is determined,and a possible magnetic structure is proposed.[25]However, this result contradicts the conventional Goodenough–Kanamori–Anderson (GKA) rules describing the magnetic interactions in insulators.[26–28]From the Raman scattering study, most phonon peaks are identified, of which several modes exhibit strong spin-phonon coupling.[28]For the magnetic scattering,three magnetic modes are observed,with the high-energy modes assigned to two-magnon modes.[28]

Up to now, very little research into the spin correlations in Ni2NbBO6is carried out,while at least two important questions still exist. One is about the complex coupling configuration, regarding the contradictory results shown above.The other is related to whether Ni2NbBO6should be treated as low-D antiferromagnets. Although this compound can be viewed as a spin chain from the structural point of view, the temperature dependence of dc-susceptibility is very similar to ordinary 3D antiferromagnets. The broad peak in the susceptibility, typical for low-D antiferromagnets,[29]is completely absent here.

With nuclei as a natural probe sitting on the lattice site,NMR is very useful in studying static magnetism and lowenergy spin excitations in the quantum antiferromagnets. In this paper, we report the study of the spin correlations in Ni2NbBO6via11B NMR spectroscopy as well as relaxation measurements. The AFM long-ranged order is indicated from the line splitting with a magnetic field applied along the b-axis.By the spectral analysis based on the lattice symmetry,this fact supports the magnetic coupling configuration previously proposed by the DFT calculations instead of the GKA rules.[25–27]From the spin-lattice relaxation rates, a prominent peak at T ∼35 K is observed, which is well above the N´eel temperature. In contrast with the temperature dependence of the Knight shift and dc-susceptibility, this peak originates from the short-ranged AFM correlations,which is the typical characteristic of quasi-1D AFM spin chains.Based on these observations, the Ni2NbBO6compound can be viewed as strongly ferromagnetically coupled armchair S=1 spin chains along the crystalline b-axis.

2. Experimental setup

Single crystals of Ni2NbBO6are synthesized with the conventional flux method as described elsewhere.[24,25]Dark green crystals with typical dimensions of 1.5×1.5×1 mm3are chosen for our NMR measurements. The crystal directions are identified by single-crystal x-ray diffraction. The precise alignment of the magnetic field with the crystalline axis is guaranteed by placing the sample on a piezoelectric nanorotation stage. Our NMR measurements are conducted on the11B nuclei (γn/2π =13.655 MHz/T, I =3/2) with a phase coherent NMR spectrometer. The11B NMR spectra are obtained by summing up the frequency-swept spin-echo intensities. The spin-lattice relaxation rates are measured by the conventional inversion-recovery pulse sequence, and fitting the nuclear magnetization to the standard recovery function for nuclei with I=3/2.

3. Results and discussion

The Ni2NbBO6crystallizes in the orthorhombic structure with the pnma space group.[24]The lattice structure is sketched in Fig.1(a). Along the crystalline b-axis, double edge-shared[NiO6]octahedra are linked by the[NbO6]octahedra and[BO4]tetrahedra,forming the armchair shaped spin chain. Alternatively,the lattice can also be viewed as a zigzag spin chain along the c-axis. Only one position of11B-site exists in the lattice.The[BO4]tetrahedra share their corners with the[NiO6]octahedra,leading to the strong coupling between the11B nuclei and the spins located on Ni2+,and making the11B-NMR a very sensitive local probe of the magnetism.

Fig.1.(a)The crystal structure of Ni2NbBO6 with distorted[NiO6]/[NbO6]octahedra and[BO4]tetrahedra shown by the respective polyhedra. (b)Typical 11B NMR spectrum at the paramagnetic state(T =140 K)with a 12 T field applied along three different crystalline axes. The red lines are fitting to the data by the Lorentz peak function.

We start to discuss our NMR results by presenting typical frequency-swept NMR spectra with the magnetic field applied along three different crystalline axes in Fig.1(b). For fields along the a- or c-axis, three resonance peaks are identified,with one central peak and double satellite peaks symmetrically located on both sides. Applying the field along the b-axis,we observe only one sharp peak. All the peaks can be well reproduced by the Lorentz peak function, as shown by the fitting lines.as[30–32]

Here K is the Knight shift defined as the relative line shift with respect to the nuclear Larmor frequency. The X,Y, Z directions are the principal axes of the electric field gradient(EFG)tensor {Vij}, and the η formulated by η ≡(VXX−VYY)/VZZis the in-plane anisotropy of the EFG. In the high magnetic field regime,the resonance frequency can be calculated by the first-order perturbation theory

where quadrupolar frequency ωQis equal to 3e2qQ/(¯h2I(2I −1)), and m is the magnetic quantum number. As a result,one can expect to observe three transitions for11B nuclei with I=3/2 for most magnetic field angles,while only one transition for some special angles(here isµ0H||b-axis).

In Fig.2,we show11B NMR spectra at different temperatures with the field applied along three different crystalline axes. With the magnetic field applied parallel with the a-or caxis(Figs.2(a)and 2(c)),the temperature dependence of11B spectra share a similar behavior. All the three peaks shift to the low frequency side with the sample cooling down for the paramagnetic state (T >TN=20 K). When the spin system enters the AFM state,the spectral frequency tends to become temperature independent.For theµ0H||b-axis case(Fig.2(b)),the single peak first shifts to the high frequency side above TN, and splits symmetrically into double peaks in the AFM ordered state. For all the three field orientations, the spectra broaden gradually upon cooling.

Fig.2. The 11B NMR spectra at different temperatures with a 12 T field applied along the crystalline a-axis(a),b-axis(b),and c-axis(c).

To study the spin susceptibility and the magnetic interactions, we plot the temperature dependence of the Knight shift and the internal field for the AFM state respectively in Figs.3(a)and 3(b). The Knight shift is a good measure of the spin susceptibility with the general expression[32]

The tensor{Aij}is the coupling constant between the nuclear and electronic spins. The second-order correction to the resonance frequency of the central transition is very small,thus is neglected in the calculation of Knight shifts. For the present sample with µ0H||a- or c- axis, the element of the coupling tensor happens to be negative, resulting in the Knight shifts with negative values. The temperature dependence of |K|(Fig.3(a))shows a typical upturn behavior upon cooling,indicating the enhancement of the spin susceptibility. This is consistent with the dc-susceptibility measurements. With the field along the b-axis,we can precisely determine the line width of the only resonance peak(shown in Fig.3(a)inset). The temperature dependence of the line width also shows an upturn behavior at low temperatures, which can be well scaled with the Knight shift. Thus,the line broadening originates from the enhanced Knight shift at low temperatures.

Next,we study the magnetic interactions via the spectral analysis below TN. Line splitting due to the setup of magnetic ordering is observed below TNwith the magnetic field applied along the b-axis. In the AFM state, the magnetic unit cell is doubled as compared with the crystalline one. One can expect a non-zero staggered internal field on the nuclear sites. When the external field is applied along the direction of the internal field, the total field can be calculated as Htot= Hext±Hint,leading to the line splitting in the AFM state.[33,34]In the present sample, the line splitting with µ0H||b-axis indicates the staggered internal field along this crystalline direction.The measured internal field serving as the order parameter of the AFM transition is plotted against temperature in Fig.3(b). By fitting the data near TNto the function Hint∝(1−T/TN)β,the critical exponent β is determined to be ∼0.35. This is consistent with the 3D characteristic of the AFM transition.[34,35]

Fig.3. (a)The temperature dependence of the Knight shift for different magnetic field orientations. The Knight shift data 11K withµ0H||a andµ0H||c is multiplied by −1 to get rid of the influence of the minus hyperfine coupling constant. Inset: The line width as a function of temperature withµ0H||b. (b)The internal field is calculated from the frequency gap of the splitting peaks compared with that at T =2 K versus temperature. (c)The simplified crystalline structure with the magnetic coupling J1, J2, and J3 marked. Only the 11B sites and the magnetic Ni2+ sites are shown for clarity. The six adjacent Ni2+ magnetic sites with respect to the 11B sites are shown on the righthand side,with the distance between them marked respectively.

Our data support the magnetic coupling configuration proposed by the DFT calculations[25]instead of the GKA rules. We show the simplified crystal structure in Fig.3,only presenting the magnetic Ni2+and11B sites. The nearestneighbour, next-nearest-neighbour, and next-next-nearestneighbour magnetic coupling are marked with J1, J2, and J3,respectively. For every11B, six adjacent Ni2+magnetic moments labeled as 1,1′,etc.,contribute to the internal field(see the enlarged version). Locally,the mirror symmetry about the plane perpendicular to the b-axis is maintained. In the magnetically ordered state,the internal field contributed by 1-sites can be generally written as[36]

where the 2-ranked tensor {Aij} describes the coupling between11B nuclear spins and the magnetic moments located on Ni2+1-sites. Under the mirror symmetry operation, the coupling tensor between11B and the 1′-sites is obtained to be

As proposed by DFT calculations,[25]the J2coupling is AFM,while J1and J3couplings are ferromagnetic.Because the magnetic frustration can be neglected in this compound,[25]the magnetic structure at low temperatures is determined by the magnetic interactions. Based on this configuration, the internal field at the11B sites contributed from the 1- and 1′-sites can be calculated as

For both 2-and 2′-sites and 3-and 3′-sites,similar results can be obtained.

From the temperature dependence of dc-susceptibility and the field induced spin-flop transition with the field along the c-axis,the ordered moment aligned with the c-axis can be easily determined in our sample,which is different from the reported a-axis.[25]However, this inconsistency does not affect the analysis here. Thus, one can imagine the staggered internal fields along the b-axis for11B nuclei stacking on different positions along the b-axis. As the Mbcomponent is zero, the line-splitting with µ0H||a-axis is absent. This is fully consistent with our NMR results. For the strong NMR magnetic field along c-axis,the line-splitting is also absent,further suggest that the magnetic moments flop to the crystalline a-axis in our sample.

Our observations contradict the magnetic coupling configuration indicated by GKA rules.[26–28]According to the GKA rules for the superexchange magnetic coupling in insulators,the sign of interaction strongly depends on the bond angle between the magnetic sites and the intermediate ligands.From the lattice structure,the J3-coupling is determined to be AFM,while the J1and J2-couplings are FM.Based on this configuration,the internal field contributed from 1-and 1′-sites is

This contradicts the line-splitting with µ0H||b-axis observed in our NMR data. The failure of GKA rules in the present sample may be related to the coupling of other side groups with the intermediate ligands,[39]which also implies the complicated magnetic coupling in the sample.

Next, we focus on the question of whether Ni2NbBO6can be viewed as low-D magnets. For the AFM chain system,a typical broad peak in the temperature dependence of the susceptibility can be expected, which results from the buildup of short-range correlations at low temperatures.[29]However, the susceptibility of Ni2NbBO6follows a well-defined Curie–Weiss upturn behavior from room temperature down to TN, similar to other ordinary 3D antiferromagnets. We study the spin excitations through the spin-lattice relaxation measurements.The spin-lattice relaxation rate(SLRR)formulated as[30,31]

In Fig.4, we present the temperature dependence of(T1T)−1for three different field directions. For µ0H||a- and c-axis,(T1)−1is obtained by fitting the nuclear magnetization recovery curve to the standard function for the 1/2 ↔−1/2 transition of the nuclei with I=3/2,[37]

For µ0H||b-axis, the single exponential magnetization recovery function is used as all the three transitions are excited in the experiment. In the present single crystal, all the fitting curves are perfect without any stretching or other different T1-components. This further indicates the uniform excitation behavior in our sample. With the sample cooling from high temperature,(T1T)−1shows an upturn behavior which can be described by the Curie–Weiss law,(T1T)−1∝1/(T+θ)(shown by the fitting lines in Fig.4). When the temperature is further lowered,(T1T)−1begins to deviate from the Curie–Weiss law,and start to drop at T*∼35 K,well above the AFM transition TN=20 K. With the sample entering the AFM long-rangeordered state, (T1T)−1shows a typical power-law temperature dependence, which is clearly shown by the dashed line in Fig.4 inset.

Fig.4. The spin-lattice relaxation rate divided by temperature (11T1T)−1 plotted versus temperature for different field directions. The solid lines are fitting to the Curie–Weiss law (see the main text). Inset: The temperature dependence of the spin-lattice relaxation rates(11T1)−1.

4. Conclusion

To conclude, we have carried out a detailed NMR study on the spin correlations in Ni2NbBO6compound. The AFM long-ranged order is monitored by the line splitting with a magnetic field along the b-axis. By the spectral analysis based on local crystalline symmetry,the magnetic coupling configuration proposed by DFT calculations is supported. The deviation from the general GKA rules is proposed to originate from the coupling of other side groups with the intermediate ligands. From the spin-lattice relaxation rates,a prominent broad peak at T*=35 K is observed in the temperature dependence of(T1T)−1.This behavior is contributed from the short-ranged AFM correlations with a typical energy scale of ∼3 meV,further indicating the low-D characteristic of the magnetic behavior in Ni2NbBO6. As a result,Ni2NbBO6can be viewed as a strongly coupled armchair spin chain system along the b-axis lying on the 1D-3D crossover regime,which has placed strong constraints on the theoretical models describing this material.