Quasi-1D XY antiferromagnet Sr2Ni(SeO3)2Cl2 at Sakai-Takahashi phase diagram

Uniform quasi-one-dimensional integer spin compounds are of interest as a potential realization of the Haldane conjecture of a gapped spin liquid. This phase, however, has to compete with magnetic anisotropy and long-range ordered phases, the implementation of which depends on the ratio of interchain J′ and intrachain J exchange interactions and both uniaxial D and rhombic E single-ion anisotropies. Strontium nickel selenite chloride, Sr2Ni(SeO3)2Cl2, is a spin-1 chain system which passes through a correlations regime at Tmax ~ 12 K to long-range order at TN = 6 K. Under external magnetic field it experiences the sequence of spin-flop at Bc1 = 9.0 T and spin-flip transitions Bc2 = 23.7 T prior to full saturation at Bsat = 31.0 T. Density functional theory provides values of the main exchange interactions and uniaxial anisotropy which corroborate the experimental findings. The values of J′/J = 0.083 and D/J = 0.357 place this compound into a hitherto unoccupied sector of the Sakai-Takahashi phase diagram.


Results
The static magnetic susceptibility χ = M/B reveals the low-dimensional nature of magnetism in Sr 2 Ni(SeO 3 ) 2 Cl 2 as well as its competition with the evolution of long-range magnetic order. Specifically, the magnetic susceptibility demonstrates a Curie-Weiss-like behavior at high temperatures, but shows a pronounced correlation hump at T max ~ 12 K before evidencing a kink at the Néel temperature T N = 6 K. The formation of long-range magnetic order is most pronounced in the Fisher specific heat ∂(χT)/∂T, as shown in the inset to Fig. 2a. The fit by the Curie-Weiss law, i.e. χ = χ 0 + C/(T − Θ), gives the Curie constant C = 1.277 ± 0.001 emu K/mol thereby defining the effective magnetic moment of the Ni 2+ ions µ eff = 3.233 ± 0.003 µ B and their g-factor g = 2.26 ± 0.01. The Weiss temperature Θ is negative, Θ = − 17.8 ± 0.1 K, which points to the predominance of antiferromagnetic exchange interactions at elevated temperatures. Finally, the temperature independent term χ 0 = − (1.82 ± 0.02) × 10 -4 emu/ mol equals the sum of Pascal constants of the constituent ions 10 .
The presence of significant antiferromagnetic correlations is evidenced by the fact that the magnetic susceptibility deviates from the Curie-Weiss behavior below about 6T N . This behavior, including the appearance of a correlation maximum, is typical for a linear spin-1 chain system. As shown in Fig. 2a, in a wide temperature range above T N the χ(T) dependence is well described by Weng equation 11 where x = J k B T , N A , µ B and k B are Avogadro, Bohr and Boltzmann constants. For fitting the static susceptibility data (solid line in Fig. 2a), both χ 0 and the g-factor were kept fixed to the abovementioned values resulting in the intrachain exchange interaction parameter J = 9.97 ± 0.02 K.
The low-dimensional nature of the spin system in Sr 2 Ni(SeO 3 ) 2 Cl 2 is further corroborated by the specific heat c p data in Fig. 2b which in addition to the anomaly at T N confirm significant entropy change well above T N . Describing the lattice specific heat by fitting a model of one Debye and three Einstein modes to the hightemperature regime from 22.5 K up to 200 K (see red line in Fig. 2b) allows obtaining the temperature dependence of the magnetic entropy S mag , as shown in Fig. 2b (right ordinate). The Debye and Einstein temperatures and respective coefficients obtained from fitting were θ D = 118.3 K, n D = 1.42, θ E,1 = 277.4 K, n E,1 = 4.55, θ E,2 = 635.9 K, n E,2 = 3.13, θ E,3 = 162.9 K, n E,3 = 2.26. The fitting procedure is explained in the methods section. The data imply that about 65% of the full magnetic entropy of Rln(3) are released below ~ 25 K while only a portion of which (~ 25%) is spent below T N . The data thereby point to significant short-range magnetic correlations well above T N and hence to the quasi one-dimensional character of magnetism in Sr 2 Ni(SeO 3 ) 2 Cl 2 .
The evolution of short-range magnetic correlations is also visible in the thermal expansion coefficient α shown in Fig. 2c. In particular, the data are well described by the phonon background at temperatures up to 55 K, which has been obtained as explained in the methods section, but exhibit a small peak at T N and a broad region of nonphononic thermal expansion, thereby implying considerable magneto-elastic coupling in Sr 2 Ni(SeO 3 ) 2 Cl 2 . Both the anomaly at T N and the length changes in the correlation regime signal the decrease of the volume of the unit cell upon evolution of short-range and long-range magnetic order, respectively. Comparing the thermal expansion coefficients with the specific heat c p allows to quantify magneto-elastic coupling by exploiting the Grüneisen ratio γ = α/c p . As shown in Fig. 2c, the magnetic contributions to the specific heat and to the thermal expansion coefficients can be scaled in the correlation regime. The data hence imply the presence of a single dominating energy scale 12 . As entropy changes are of magnetic nature, we conclude that a single magnetic degree of freedom, i.e., the magnetic intrachain exchange J, drives the observed non-phonon length and entropy changes. The corresponding scaling parameter yields the hydrostatic pressure dependence of J via the Grüneisen equation 13 γ = α mag /c p,mag = 1/V m ∂lnJ/∂p| T = 0.24(2) mol/MJ, with V m = 1.26 10 -4 m 3 /mol being the molar volume, calculated from the unit cell volume V cell = 417.66 Å 314 . However, the Grüneisen ratio changes at the λ-like anomaly around www.nature.com/scientificreports/ T N and reaches γ = 0.16(2) mol/MJ. We conclude that the formation of long-range magnetic order at T N is either driven by more than one energy scale or an energy different from J. This is expected as the evolution of long-range magnetic order appears to result only in the presence of J′ and is affected by anisotropy D. Quantitatively, using the Ehrenfest relation yields the hydrostatic pressure dependence dT N /dp = T N V m γ = 0.12(4) K/GPa.

Electronic structure and exchange interactions
The electronic properties including exchange interactions for Sr 2 Ni(SeO 3 ) 2 Cl 2 have been calculated in the frame of the GGA + U approach.  In order to identify the origin of the low-dimensional magnetic behavior of Sr 2 Ni(SeO 3 ) 2 Cl 2 the calculations of Heisenberg exchange parameters using the total energy method implemented in the JaSS code 15 have been performed. To obtain the values of the main exchange parameters, shown by arcs in Fig. 1, DFT + U calculations for the supercell, containing 8 Ni 2+ ions of 4 magnetic configurations, were carried out. The Heisenberg model was chosen to be with J > 0 corresponding to antiferromagnetic exchange. The sum runs once over each pair of magnetic ions, numerated by i,j indices for intrachain interactions and i,k indices for interchain interactions. The calculations yield J = 9.75 K-the intrachain exchange interaction parameter between Ni 2+ ions located at a distance of 5.325(1) Å along the a-axis, J′ = 0.8 K-exchange interaction parameter at a distance of 6.436(1) Å along the b axis, and J″ = − 0.35 K-exchange interaction parameter corresponding to a distance of 7.294(1) Å in the ac-plane. At these parameters, one might expect a long-range magnetic order with Néel temperature where critical value of interchain interaction J ′ c ∼ Je −π which promotes the formation of true long-range order 16 . |J′|+ 2|J″|= 1.5 K yields T N ~ 4.8 K. The single-ion anisotropy may influence this value. Calculating in the GGA + U + SOC approximation the total energies of different configurations with spins lying in the (oxygen) basal plane and along Ni-Cl bonds we find that the single ion anisotropy is of easy-plane type, D = 3.5 K.

Discussion
The analysis of the static magnetic susceptibility, the specific heat and the thermal expansion clearly confirms the one-dimensional nature of magnetism in Sr 2 Ni(SeO 3 ) 2 Cl 2 governing the properties above T N and a separated regime of long-range magnetic order at lowest temperatures. The numerical results agree with the one-dimensional nature of magnetism in this compound. The numerically obtained nearest-neighbor intrachain coupling agrees quite well with the experimentally determined value. Further information on the magnetic parameters is provided by studies of the field dependence of the magnetization, at low temperatures, as shown in Fig. 4. The behavior at lowest magnetic fields, i.e. the convex curvature, is influenced by the presence of about 1% of defects/ impurities. At higher fields, the M(B) curve evidences a sequence of phase transitions at B C1 = 9 T and B C2 = 23.7 T while at B sat = 31.0 T the magnetization arrives to the full saturation at M sat = 2.15 µ B .
While the polycrystalline nature of the material under study may suggest the presence of two features at high magnetic fields associated with the saturation fields for the different directions of the field with respect to the g-factor, the difference in B c2 and B sat is too large for a reasonable anisotropy of the g-factor. In addition, the presence of a low-field feature is unexpected for an easy-plane magnet. The results shown in Fig. 4, however, clearly rule out an easy-axis antiferromagnet where in case of not too large anisotropy the sequence of a spin-flop (at B c1 ) and a spin-flip transition (at B c2 ) fully describe the evolution of magnetization under magnetic field. Evidently, this is not the case for Sr 2 Ni(SeO 3 ) 2 Cl 2 as the magnetization rises significantly till saturation at B sat > B c2 . In contrast, for an easy-plane antiferromagnet there should be no spin-flop transition, since for any direction of external magnetic field the magnetic moments will orient perpendicular to the field. We hence conclude the presence of The analysis of the M versus B curve hence yields an additional set of magnetic parameters which not only agree to the value of J deduced from the analysis of the static susceptibility, but also agrees well to J and D obtained numerically from DFT. Both experimental and calculated values of J, D and E, as well as critical fields B c1 , B c2 and B sat are given in Table 1.
In    and NiSb 2 O 6 28 . Judging from the claimed values of J′/J and D/J ratios both systems should fall into the Haldane sector of this diagram. However, opposite to these estimations both nickel tantalate and nickel antimonite reach long-range order at high enough temperatures T N = 10.3 K and 6.7 K, respectively. The title compound, Sr 2 Ni(SeO 3 ) 2 Cl 2 , is unequivocally positioned at J′/J > 0 and 0 < D/J < 1 in the hitherto unoccupied easy-plane antiferromagnetic (XY-AFM) long-range ordered sector of the Sakai-Takahashi phase diagram. This makes Sr 2 Ni(SeO 3 ) 2 Cl 2 an extremely interesting compound both for further theoretical and experimental studies. Varying the alkaline-earth metal or applying external pressure or strain one obtains a playground for studying yet unexplored region of the phase diagram and gaining deeper insight into Haldane physics.

Methods
The sample of strontium nickel selenite chloride was obtained from SrSeO 3 and anhydrous NiCl 2 precursors 29 . The mixture of 0.2 g SrSeO 3 and 0.06 g of NiCl 2 has been ground in the agate mortar and loaded into a quartz tube. The tube was sealed under vacuum and placed into the furnace for 10 days at 750 °C. The X-ray pattern was fully indexed in the monoclinic unit cell with parameters a = 5.3373(21)Å, b = 6.4526(23) Å, c = 12.231(4) Å, β = 92.518(20)°, space group P2 1 /n 30 .
Thermodynamic properties, i.e. magnetization M and specific heat C p , of the pressed pellet sample were studied using various options of Magnetic Properties Measurements System MPMS-7 T and Physical Properties Measurements System PPMS-14 T (Quantum Design) up to 14 T in the temperature range from 2 to 300 K. Pulsed-magnetic-field magnetization has been measured up to 45 T using a coaxial pick-up coil system. The pulsed-field magnetization data have been calibrated by means of static magnetic field measurements. The measurements of thermal expansion have been provided by means of a three-terminal high-resolution capacitance dilatometer in a home-built set-up 31 . The final fitting procedure, after trying several ways of approaching the fitting, for the phonon background was the following: first, the thermal expansion data up to 55 K, excluding the range from 3 to 22.5 K was fitted with one Debye and two Einstein modes. This yielded θ D = 118.3 K, θ E,1 = 277.4 K and θ E,2 = 1497.81 K. θ E,2 only contributes to the overall thermal expansion above 70 K. Therefore, the thermal expansion data was re-fit with one Debye mode and one Einstein mode with fixed θ D = 118.3 K, θ E = 277.4 K, which corresponds to the background shown in Fig. 2 and used to obtain α mag . To satisfactorily describe the specific heat data, two Einstein modes were then added and fit with fixed θ D = 118.3 K, θ E,1 = 277.4 K and freely varying θ E,2 and θ E,3 . This fit resulted in the final parameters as described in the text. Lastly, the fit was subtracted from c p to obtain the magnetic contribution c p,mag to the specific heat.
To calculate the electronic and magnetic structure of Sr 2 Ni(SeO 3 ) 2 Cl 2 within the density functional theory the pseudopotential VASP code 32 with the Perdew-Burke-Ernzerhof version of the exchange correlation potential 33 has been used. To take into account the strong electronic correlations on Ni the GGA + U approach has been applied 34 . On-site Hubbard U and intra-atomic exchange J H for Ni 2+ ions were chosen to be 8.0 and 0.9 eV, respectively 35,36 . A 5 × 3 × 3 k-mesh in the symmetry-irreducible part of the first Brillouin zone was used in all calculations. The convergence condition for the total energy was set to 10 -6 eV.