Quasi two-dimensional magnetism in spin-$\frac{1}{2}$ square lattice compound Cu[C₆H₂(COO)₄][H₃N-(CH₂)₂-NH₃]$\cdot$3H₂O

Quasi-two-dimensional (2D) antiferromagnets are ideal materials to study the interplay between quantum fluctuations and magnetic frustration due to competing interactions. Frustrated square lattice (FSL or $J_{1}-J_{2}$ model) model is the best known example in this category. The Hamiltonian of the isotropic FSL model can be written as

where $J_{1}$ and $J_{2}$ are the nearest-neighbour (NN) (along the edge) and next-nearest-neighbour (NNN) (along the diagonal) interactions, respectively in a square. The classically possible ground states in this model are determined by the frustration angle $\phi=\text{tan}^{-1}(J_{2}/J_{1})$.Shannon et al. (2004) There are three possible order states: Néel antiferromagnetic (NAF, $-0.5\pi\leq\phi\leq 0.15\pi$), columnar antiferromagnetic (CAF, $0.15\pi\leq\phi\leq 0.85\pi$), and ferromagnetic (FM, $0.85\pi\leq\phi\leq-0.5\pi$) states with wave vectors ($Q_{x}$, $Q_{y}$) = ($\pi$, $\pi$), [($\pi$, $0$) or ($0$, $\pi$)], and ($0$, $0$), respectively.Shannon et al. (2006); Schmidt and Thalmeier (2017) The transition regimes NAF/CAF and CAF/FM are known as quantum critical regimes, though the precise boundaries of these regimes are not yet well defined. It is proposed that the ground state in these critical regimes are not exactly quantum spin-liquid but different dimer phases with a singlet gap and gapless nematic phases, respectively.Sushkov et al. (2001); Shannon et al. (2006); Singh et al. (1999); Capriotti et al. (2003); Schmidt et al. (2007a, b)

The $J_{1}-J_{2}$ phase diagram has been extended further to the spatially anisotropic square lattice or rectangular lattice (known as $J_{1a,b}-J_{2}$ model).Schmidt et al. (2011a) The Hamiltonian for a 2D $S=1/2$ frustrated rectangular lattice (FRL) model can be written as

Here, $J_{\rm 1a}$ and $J_{\rm 1b}$ are the anisotropic exchange couplings along the edges of the square and the coupling along the diagonals ($J_{2}$) remains same. The classically predicted phase diagram becomes a function of frustration angle $\phi=\text{tan}^{-1}\left(J_{2}/\sqrt{\frac{(J_{1a}^{2}+J_{1b}^{2})}{2}}\right)$ and anisotropy parameter $\theta=\text{tan}$${}^{-1}(J_{1b}/J_{1a})$. The introduction of a rectangular distortion does not significantly change the phase diagram. The predicted phases are FM, NAF, and columnar antiferromagnets [CAF_a, ($Q_{x}$, $Q_{y}$) = ($\pi$, 0) and CAF_b, ($Q_{x}$, $Q_{y}$) = (0, $\pi$)]. The only difference is that the CAF phases are degenerate for the isotropic model $(J_{1a}=J_{1b})$ with $\theta=\frac{\pi}{4}$ or $\theta=\frac{-3\pi}{4}$. Further, the $J_{1a,b}-J_{2}$ model predicts that the CAF phase is stable for all values of $\phi$, especially in the spin nematic phase regime of the isotropic $J_{1}-J_{2}$ model.Schmidt et al. (2011a)

The $S=1/2$ FSL model has been realized in the class of layered V⁴⁺ based inorganic compounds $AA^{\prime}$(VO)(PO₄)₂ ($AA^{\prime}$ = Zn₂, Pb₂, SrZn, PbZn, BaZn, and BaCd) and Li₂VO(Si,Ge)O₄.Nath et al. (2008, 2009); Yogi et al. (2015); Tsirlin et al. (2010); Carretta et al. (2009); Bossoni et al. (2011); Nath et al. (2009); Roy et al. (2011); Nath et al. (2008); Tsirlin et al. (2010); Rosner et al. (2003); Bettler et al. (2019); Tsirlin et al. (2011, 2009) Among these compounds, BaCdVO(PO₄)₂ is the one located very close to the nematic phase regime in the $J_{1}-J_{2}$ phase diagram and is being extensively studied. Some of the recent studies have reported the signature of spin nematic phase in BaCdVO(PO₄)₂.Povarov et al. (2019); Skoulatos et al. (2019); Bhartiya et al. (2019) A few metal-organic compounds based on V⁴⁺ and Cu²⁺ have also been studied in light of the 2D spin-$1/2$ Heisenberg model.Guchhait et al. (2019); Woodward et al. (2002); Rønnow et al. (2001); Nath et al. (2015) A series of Cu based quasi-2D organometallic magnets where Cu²⁺ ions are bridged by pyrazine molecules are [Cu(HF₂)(pyz)₂]$X$ ($X$ = BF${}_{4}^{-}$, ClO${}_{4}^{-}$, PF${}_{6}^{-}$, SbF${}_{6}^{-}$, and AsF${}_{6}^{-}$)Goddard et al. (2008) and [Cu(pyz)₂]$X_{2}$ ($X$= ClO${}_{4}^{-}$ and BF${}_{4}^{-}$).Lancaster et al. (2007); Woodward et al. (2007); Tsyrulin et al. (2010) These compounds are having square lattice network with negligible NNN exchange coupling ($J_{2}$). Another family of Cu based organo-metallic square lattice compounds are $A_{2}$Cu$X_{4}$ ($A$ = 5CAP and 5MAP, $X$ = Br and Cl) without frustration.Woodward et al. (2002) Recently, we have reported that Cu[C₆H₂(COO)₄][C₂H₅NH₃]₂ is a quasi-2D spatially anisotropic non-frustrated spin-$1/2$ square lattice with exchange couplings $J_{1a}/k_{\rm B}=5.6$ K and $J_{1c}/k_{\rm B}=8.0$ K along $a$- and $c$-directions, respectively.Nath et al. (2015)

In this work, we report the synthesis and magnetic properties of a new organic spin-1/2 quantum magnet Cu[C₆H₂(COO)₄][H₃N-(CH₂)₂-NH₃]$\cdot$3H₂O (or C₁₂H₁₈CuN₂O₁₁). The magnetization data analysis confirms the non-frustrated quasi-2D nature with a weak anisotropy in the in-plane couplings. It does not show the onset of magnetic long-range-ordering (LRO) down to 2 K, reflecting weak inter-plane coupling and hence perfect two-dimensionality.

Single crystals of the Cu(II)-based metal organic hybrid compound C₁₂H₁₈CuN₂O₁₁ were synthesised by using 1,2,4,5-benzenetetracarboxylic acid (H₄BTC). Since the compound contains four carboxylic acid groups, we were initially getting mixture of products from which isolation of pure phase of the material was difficult. After repeated trials and by varying the reaction conditions the phase-pure form of the compound was obtained by adopting the following procedure. Copper acetate monohydrate (5 mmol, 1.00 g), ethylene diamine (5 mmol, 0.35 mL), H₄BTC (5 mmol, 1.27 g) were reacted in 30 mL DMF-water mixture (taken in 1:1 volume ratio). The initial blue product formed was filtered out. The clear and pale blue filtrate obtained was kept for slow evaporation for 8 days at room temperature. Light bluish needle type crystals of the target compound in phase-pure form were separated and dried in air. The yield was 45% (based on Cu).

Single crystal x-ray diffraction (XRD) was performed on a good-quality single crystal at room temperature using a Bruker KAPPA APEX-II CCD diffractometer equipped with graphite monochromated Mo $K_{\alpha 1}$ radiation ($\lambda=0.71073$ Å). The data were collected using APEX3 software and reduced with SAINT/XPREP.Bruker (2016) An empirical absorption correction was done using the SADABS program.Sheldrick (1994) The structure was solved with direct methods using SHELXT-2018/2Sheldrick (2015) and refined by the full matrix least squares on $F^{2}$ using SHELXL-2018/3, respectively.Sheldrick (2018) All the hydrogen atoms were placed geometrically and held in the riding mode for the final refinements. The final refinements included atomic positions for all the atoms, anisotropic thermal parameters for all the nonhydrogen atoms, and isotropic thermal parameters for the hydrogen atoms. The crystal data and details of the structure refinement parameters are listed in Table 1.

As the size of the crystals was too small, it was not possible to do the magnetic measurements on the individual crystals and hence powder sample was used for this purpose. The temperature ($T$) dependent magnetic susceptibility [$\chi(T)$] in four different magnetic fields ($\mu_{\rm 0}H=0.5$, 1, 3, and 5 T) was measured in the temperature range $2\leq T\leq 300$ K using the vibrating sample magnetometer (VSM) attachment to the Physical Property Measurement System (PPMS, Quantum Design). A magnetic isotherm (magnetization $M$ vs field $H$) was measured by varying the magnetic field from 0 to 14 T at $T=1.9$ K.

The Quantum Monte Carlo (QMC) simulation for magnetization was performed assuming the Heisenberg model on a nonfrustrated square lattice with an isotropic exchange coupling. We used the Hamiltonian in the presence of a magnetic field $\hat{\mathcal{H}}=J\sum_{\langle i,j\rangle}S_{i}\cdot S_{j}-H\sum_{i}S^{z}_{i}$, where $J$ represents the exchange coupling strength between spins at the $i^{th}$ and $j^{th}$ sites and $H$ is the external magnetic field. We used the directed loop QMC algorithm in the stochastic series expansion representationSandvik (1999); Alet et al. (2005) implemented in the ALPS software package.ALP The lattice size was taken to be $20\times 20$ (400 sites) and measurements were done from a simulation of about $10^{5}$ sweeps including about 5000 thermalization sweeps.

According to mean field approximation, for the FSL model, one can write $\theta_{\rm CW}=\frac{zS(S+1)}{3k_{\rm B}}(J_{1}+J_{2})$.Domb et al. (1964) Taking $S=1/2$, $z=4$, $J_{1}/k_{\rm B}\simeq 5.35$ K, and $J_{2}/k_{\rm B}\simeq-0.01$ K, we got $\theta_{\rm CW}\simeq 5.34$ K which is very close to the CW temperature obtained from the $1/\chi$ analysis. Using the values of $J_{1}$ and $J_{2}$, the frustration control parameter is calculated to be $\phi=-0.1^{\circ}$ ($\sim-0.0006\pi$), which places the compound in the NAF ordered state of the $J_{1}-J_{2}$ phase diagram.Nath et al. (2008) Similarly, for a FRL model one can write $\theta_{\rm CW}=(\frac{J_{\rm 1a}+J_{\rm 1b}}{2}+J_{\rm 2})/k_{\rm B}$. Taking $J_{\rm 1a}/k_{\rm B}\simeq 5.31$ K, $J_{\rm 1b}/k_{\rm B}\simeq 5.38$ K, and $J_{2}/k_{\rm B}\simeq-0.24$ K we got $\theta_{\rm CW}\simeq 5.11$ K which is even closer to the CW temperature obtained from the $1/\chi$ analysis. The anisotropic angle and frustration angle are estimated to be $\theta\simeq 0.252\pi$ and $\phi\simeq-0.014\pi$, respectively in the NAF regime of the $J_{1a,b}-J_{2}$ phase diagram.Schmidt et al. (2011a)

Usually, in a frustrated magnet, the extent of frustration can be quantified by the frustration parameter $f=\frac{|\theta_{\rm CW}|}{T_{\rm N}}$. C₁₂H₁₈CuN₂O₁₁ has no magnetic LRO down to 2 K which makes this system a good example of a quasi-2D AF system. The lower limit of the frustration parameter of this compound is estimated to be $f>\frac{5.17}{2}\simeq 2.6$, taking the upper limit of $T_{\rm N}=2$ K. Here, $|\theta_{\rm CW}|>T_{\rm N}$ implies that the magnetic LRO ($T_{\rm N}$) is prevented by quantum fluctuations due to low dimensionality of the spin-lattice and the role of frustration has negligible effect. Further, assuming that $T_{\rm N}<2$ K and using the appropriate exchange couplings, the upper limit of the inter-layer coupling is estimated to be negligibly small compared to the intra-layer coupling.Majlis et al. (1992); Schmidt and Thalmeier (2017) Thus, this compound is another example of a quasi-2D nonfrustrated system with $J_{1}/T_{\rm N}>2.67$, similar to the compounds tabulated in Ref. Guchhait et al. (2019).

From the crystal structure, the Cu-Cu distance along $b$-direction is greater than the one along $a$-direction. Therefore, one would expect $J_{\rm 1a}$ to be larger than $J_{\rm 1b}$. Similar scenario has been realized in Cu[C₆H₂(COO)₄][C₂H₅NH₃]₂ in which the DFT calculations show that $J_{\rm 1a}<J_{\rm 1c}$, even though the Cu-Cu distance along $a$-direction is alomost half of the distance along $c$-direction.Nath et al. (2015) This non-trivial behaviour is attributed to the characteristic features of [C₆H₂(COO)₄]^4- anion through which the superexchange takes place. In Cu[C₆H₂(COO)₄][C₂H₅NH₃]₂, the effective bridging angles between C atoms belonging to the C₆-phenyl ring along the superexchange paths are $\angle\psi\simeq 59.9^{\circ}$ and $\angle\xi\simeq 120.1^{\circ}$ for $J_{\rm 1a}$ and $J_{\rm 1c}$, respectively in the $ac$-plane. Therefore, it is argued that according to Goodenough-Kanamori-Anderson rules one finds $J_{\rm 1c}>J_{\rm 1a}$ and does not follow Cu-Cu distance. As shown in Fig. 4, in C₁₂H₁₈CuN₂O₁₁, the angles are $\angle\psi\simeq 60.5^{\circ}$ and $\angle\xi\simeq 119.6^{\circ}$. This explains why $J_{\rm 1a}$ and $J_{\rm 1b}$ have nearly equal values despite different Cu-Cu distances. However, to establish this proposition, a precise estimation of exchange couplings using band structure calculation is required.

In summary, we have synthesized single crystals of C₁₂H₁₈CuN₂O₁₁ and reported its crystal structure and magnetic properties in detail. C₁₂H₁₈CuN₂O₁₁ crystallizes in a monoclinic crystal structure with space group $C2/m$. Because of the low symmetry crystal structure, Cu²⁺ ions form anisotropic square lattices. The analysis of $\chi(T)$ demonstrates that the compound behaves as a nearly nonfrustrated spin-$1/2$ square lattice with $J_{1}/k_{\rm B}\simeq 5.3$ K, despite its anisotropic (or rectangular) structural arrangement. Further, the shape of the magnetic isotherm at $T=1.9$ K could be reproduced well by the QMC simulation assuming a non-frustrated square lattice with $J/k_{\rm B}=5.2$ K, supporting the $\chi(T)$ analysis. No sign of magnetic LRO down to 2 K indicates minuscule inter-plane coupling in the system. In this compound $J_{2}$ is negligibly small, but its strength can be increased and the frustration ratio $J_{2}/J_{1}$ can be tuned by an appropriate choice of the organic ligand that provides superexchange pathway between the magnetic ions. Thus, the metal organic complexes can reciprocate the inorganic compounds as the model systems in the $J_{1}-J_{2}$ phase diagram.

SG and RN acknowledge SERB, India for financial support bearing sanction order no. CRG/2019/000960. We thank Alex Andrews for his help in solving the crystal structure.