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

Abstract

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 T_max ~ 12 K to long-range order at T_N = 6 K. Under external magnetic field it experiences the sequence of spin-flop at B_c1 = 9.0 T and spin-flip transitions B_c2 = 23.7 T prior to full saturation at B_sat = 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.

Introduction

Among the basic concepts stimulating research on strongly correlated electron systems is the Haldane conjecture of a gapped spin liquid implemented in the uniform chains of Heisenberg integer spins^1,2. Tens of compounds, both organic and inorganic, have been studied to verify this idea and some of them have been identified as belonging to the Haldane sector of the Sakai-Takahashi phase diagram³. This representation takes into account the ratios between interchain J′ and intrachain J exchange interactions and uniaxial D single-ion anisotropy⁴. In addition, properties of uniform integer spin chains can be influenced by rhombic single-ion anisotropy E⁵ and anisotropy of spin–spin exchange interactions⁶.

In the Sakai-Takahashi phase diagram, J′/J versus D/J, the existence of the Haldane phase is limited by the critical value of J′/J = 0.0162 for the square lattice of adjacent chains⁷. At positive D, the Haldane sector is restricted by D/J = 0.968⁸, beyond which the large-D, or quantum paramagnet, phase continues. At negative D, the critical point at D/J = − 0.316 distinguishes the Haldane and Néel phases⁹.

Both the presence of quantum fluctuations—which are quite pronounced for low-dimensional low-spin systems—and of frustration of interchain exchange interactions suppress the Néel phase. On the contrary, the competition between nearest-neighbor and next-nearest-neighbor exchange interactions within the chains restricts the Haldane phase. Basically, the identification of the quantum ground state of a given magnetic system is possible both experimentally and theoretically. The case of a uniform quasi-one-dimensional spin S = 1 system, strontium nickel selenite chloride, Sr₂Ni(SeO₃)₂Cl₂, illustrates the remarkable correspondence of these two approaches, which places this compound into the yet unoccupied sector of the Sakai-Takahashi phase diagram, i.e. the sector of an easy-plane XY antiferromagnet at D < 1.

In the structure of Sr₂Ni(SeO₃)₂Cl₂, the Ni²⁺ ions are positioned at the center of octahedra formed by four oxygen ions in the basal plane and two chlorine ions in apical positions. The SeO₃ pyramids bridge the NiO₄Cl₂ octahedra into uniform spin S = 1 chains running along the a-axis. The crystal structure of Sr₂Ni(SeO₃)₂Cl₂ in polyhedral representation is shown in Fig. 1.

Figure 1

The crystal structure of Sr₂Ni(SeO₃)₂Cl₂. Strontium ions are shown by blue spheres, nickel ions in black. The solid line marks the unit cell. The arcs mark exchange interaction pathways (see the text). VESTA software has been used for crystal structure visualization³⁷.

Results

The static magnetic susceptibility χ = M/B reveals the low-dimensional nature of magnetism in Sr₂Ni(SeO₃)₂Cl₂ 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 = χ₀ + C/(T − Θ), gives the Curie constant C = 1.277 ± 0.001 emu K/mol thereby defining the effective magnetic moment of the Ni²⁺ 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 χ₀ = − (1.82 ± 0.02) × 10^–4 emu/mol equals the sum of Pascal constants of the constituent ions¹⁰.

Figure 2

Temperature dependence of the static magnetic susceptibility χ = M/B taken at B = 0.1 T. The solid line represents fit by the Weng Eq. (1). The Fisher specific heat ∂(χT)/∂T is shown in the inset (a). Temperature dependences of specific heat c_p/T (left ordinate) and magnetic entropy S_mag (right ordinate). The estimated phonon contribution is shown by the solid red line (b). Temperature dependences of the magnetic contributions to the specific heat c_p,mag and thermal expansion α_mag. The inset shows α as measured and the phonon background (red line) (c).

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 equation¹¹

$$\frac{\chi {k}_{B}T}{{N}_{A}{(g{\mu }_{B})}^{2}}=\frac{2+0.0194x+0.777{x}^{2}}{3+4.346x+3.232{x}^{2}+5.834{x}^{3}}$$

(1)

where $x=\frac{J}{{k}_{B}T}$, 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 χ₀ 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 Sr₂Ni(SeO₃)₂Cl₂ 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 high-temperature 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 Sr₂Ni(SeO₃)₂Cl₂.

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 non-phononic thermal expansion, thereby implying considerable magneto-elastic coupling in Sr₂Ni(SeO₃)₂Cl₂. 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 scale¹². 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 equation¹³ γ = α_mag/c_p,mag = 1/V_m ∂lnJ/∂p|_T = 0.24(2) mol/MJ, with V_m = 1.26 10^–4 m³/mol being the molar volume, calculated from the unit cell volume V_cell = 417.66 ų¹⁴. However, the Grüneisen ratio changes at the λ-like anomaly around 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_NV_mγ = 0.12(4) K/GPa.

Electronic structure and exchange interactions

The electronic properties including exchange interactions for Sr₂Ni(SeO₃)₂Cl₂ have been calculated in the frame of the GGA + U approach. Figure 3 represents total and partial densities of states in Sr₂Ni(SeO₃)₂Cl₂. As in other charge-transfer insulators the top of the valence band is formed mostly by O 2p and Cl 3p states, while the bottom of the conduction band is dominated by Ni 3d states. The band gap is about 2 eV. The valence band width is about 9 eV and there is a strong hybridization of Ni 3d, O 2p and Cl 3p states. The conduction band width is about 5.5 eV.

Figure 3

Total and partial density of states (DOS) plots in Sr₂Ni(SeO₃)₂Cl₂. The Fermi level has been shifted to zero.

In order to identify the origin of the low-dimensional magnetic behavior of Sr₂Ni(SeO₃)₂Cl₂ the calculations of Heisenberg exchange parameters using the total energy method implemented in the JaSS code¹⁵ 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 Ni²⁺ ions of 4 magnetic configurations, were carried out. The Heisenberg model was chosen to be

$$H = \Sigma_{{{\text{i}} > {\text{j}}}} JS_{{\text{i}}} S_{{\text{j}}} + \Sigma_{{{\text{i}} > {\text{k}}}} J^{\prime } S_{{\text{i}}} S_{{\text{k}}} + \Sigma_{{\text{i}}} D\left( {S_{i}^{{\text{z}}} } \right)^{{2}} ,$$

(2)

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 Ni²⁺ 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

$${T}_{N}\propto \mathit{ln}\left|\frac{{J}^{^{\prime}}}{{J}_{c}^{^{\prime}}}\right|\sqrt{\frac{{J}^{^{\prime}}}{J}}J$$

(3)

where critical value of interchain interaction ${J}_{c}^{^{\prime}}\sim {Je}^{-\pi }$ which promotes the formation of true long-range order¹⁶. |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 Sr₂Ni(SeO₃)₂Cl₂ 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.

Figure 4

Magnetic field dependence of the magnetization M (black filled markers) in the upsweep mode and the magnetic susceptibility ∂M/∂B (blue open markers) obtained in pulsed magnetic fields up to 45 T at T = 1.46 K. The data have been normalized by quasi-static measurements up to 14 T. No appreciable magnetocaloric effect has been noticed. The sequence of critical fields at B_c1, B_c2 and B_sat can be treated as follows¹⁷.

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 Sr₂Ni(SeO₃)₂Cl₂ 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 rhombic anisotropy E within the XY plane along with uniaxial anisotropy D along the Z axis in order to account for the observed features. In this case, the sequence of features at B_c1, B_c2 and B_sat can be treated as follows.

The spin-flop transition within the XY plane takes place at

$${B}_{c1}=\sqrt{2EJ},$$

(4)

the spin-flip transition within the XY plane occurs at

$${B}_{c2}=2J,$$

(5)

and, finally, the full saturation along the Z axis takes place at

$${B}_{sat}=2J+D.$$

(6)

Experimentally found critical values in the M(B) curve (in Tesla) can be put into correspondence with DFT results (in Kelvin) at the field of the spin-flip transition¹⁷

$$g{\mu }_{B}{B}_{c2}=2S{k}_{B}J$$

(7)

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.

Table 1 The magnetic subsystem parameters and critical fields in Sr₂Ni(SeO₃)₂Cl₂. The uncertainties in experimental data are assumed to be one unit in the last quoted digit.

In the original calculations of Sakai and Takahashi within the mean-field approximation (MFA) the critical value limiting the Haldane phase at D = 0 is J′/J = 0.013 for the square lattice of adjacent chains and 0.017 for the triangular or kagome lattices⁴. Quantum Monte Carlo calculations^7,8 provide slightly higher values 0.0162 and 0.0219 for the square and triangular lattices, respectively. As soon as the critical ratio J′/J depends on the geometry of adjacent chains, for accurate comparison of the systems with different geometry it is preferable to use the quantity zJ′/J, where z is the chain coordination number, or more complex equations if different interchain exchange interactions are present. A lower bound for zJ′/J is 0.051 in MFA and 0.068 in QMC⁸. In Sr₂Ni(SeO₃)₂Cl₂, we find (J′ + 2 J″)/J = 0.154 and D/J = 0.54 (exp) and D/J = 0.36 (DFT) which place this system into a yet unexplored and rather unique position in the Sakai-Takahashi phase diagram, as shown in Fig. 5.

Figure 5

Sakai-Takahashi phase diagram for uniform spin-1 chain compounds. Sharp borders between various phases are obtained within the mean-field approximation, while the comb follows from the quantum Monte Carlo calculations.

Conclusion

The Haldane conjecture on gapped integer spin liquids belongs to the milestones of low-dimensional quantum magnetism¹⁸. Up to date, several inorganic compounds, i.e. Y₂BaNiO₅¹⁹, AgVP₂S₆²⁰, PbNi₂V₂O₈²¹ and after some controversy SrNi₂V₂O₈^22,23, were put into the Haldane sector of the Sakai-Takahashi phase diagram. Few more, CsNiCl₃²⁴, RbNiCl₃²⁵ and LiVSi₂O₆²⁶ were assigned to the easy-axis antiferromagnetic sector (Z—AFM) of this diagram. Contradictory results have been reported for two trirutile-type tetragonal compounds, NiTa₂O₆²⁷ and NiSb₂O₆²⁸. 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, Sr₂Ni(SeO₃)₂Cl₂, 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 Sr₂Ni(SeO₃)₂Cl₂ 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 SrSeO₃ and anhydrous NiCl₂ precursors²⁹. The mixture of 0.2 g SrSeO₃ and 0.06 g of NiCl₂ 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 P2₁/n³⁰.

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-up³¹. 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 Sr₂Ni(SeO₃)₂Cl₂ within the density functional theory the pseudopotential VASP code³² with the Perdew-Burke-Ernzerhof version of the exchange correlation potential³³ has been used. To take into account the strong electronic correlations on Ni the GGA + U approach has been applied³⁴. On-site Hubbard U and intra-atomic exchange J_H for Ni²⁺ 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.