Singlet Ground State and Spin Gap in S=1/2 Kagome Antiferromagnet Rb$_2$Cu$_3$SnF$_{12}$

We investigated the crystal structure of Rb$_2$Cu$_3$SnF$_{12}$ and its magnetic properties using single crystals. This compound is composed of Kagome layers of corner-sharing CuF$_{6}$ octahedra with a 2a x 2a enlarged cell as compared with the proper Kagome layer. Rb$_2$Cu$_3$SnF$_{12}$ is magnetically described as an $S$=1/2 modified Kagome antiferromagnet with four kinds of neighboring exchange interaction. From magnetic susceptibility and high-field magnetization measurements, it was found that the ground state is a disordered singlet with the spin gap, as predicted from a recent theory. Exact diagonalization for a 12-site Kagome cluster was performed to analyze the magnetic susceptibility, and individual exchange interactions were evaluated.

Antiferromagnets on highly frustrated lattices produce a rich variety of physics. 1,2 In particular, a twodimensional Heisenberg Kagomé antiferromagnet (2D HKAF) is of great interest from the viewpoint of the interplay of the frustration and quantum effects. There are many theoretical studies on the 2D HKAF. The spin wave theory for a large spin value predicted an ordered ground state with the so-called √ 3 × √ 3 structure, which is selected by quantum fluctuation from infinite classical ground states, 3,4 whereas for a small spin value, a disordered ground state was observed by various approaches. [5][6][7][8][9] Recent careful analyses and numerical calculations for an S=1/2 case demonstrated that the ground state is a spin liquid state composed of singlet dimers only, and that the ground state is gapped for triplet excitations, but gapless for singlet excitations. [10][11][12] Consequently, magnetic susceptibility has a rounded maximum at T ∼(1/6)J/k B and decreases exponentially toward zero with decreasing temperature, while specific heat exhibits a power law behavior at low temperatures. 8,13 Specific heat also shows an additional structure, peak or shoulder at low temperatures after exhibiting a broad maximum at T ∼(2/3)J/k B .
The experimental studies of the S=1/2 HKAF have been limited, and the above-mentioned intriguing predictions have not been verified experimentally. The cupric compounds Cu 3 V 2 O 7 (OH) 2 18 Recently, the herbertsmithite ZnCu 3 (OH) 6 Cl 2 with the proper Kagomé lattice has attracted considerable attention. [19][20][21][22][23][24][25] Although no magnetic ordering occurs down to 50 mK, 20 magnetic susceptibility exhibits a rapid increase at low temperatures. 21,22 This behavior was ascribed to a large number of idle spins (4∼10 %) produced by intersite mixing between Cu 2+ and Zn 2+ 22, 23, 25 and/or the Dzyaloshinsky-Moriya interaction. 24 The singlet ground state has not been observed in these systems. The search for new 2D HKAFs with S=1/2 continues. Cs 2 Cu 3 ZrF 12 and Cs 2 Cu 3 HfF 12 have the proper Kagomé layer at room temperature 26 and are promising S=1/2 HKAFs. 27 Unfortunately, these systems undergo structural phase transitions at T t = 220 and 170 K, respectively, and also magnetic phase transitions at T N ≃ 24 K. 27 However, the magnetic susceptibilities observed at T > T t can be perfectly described using theoretical results for an S = 1/2 HKAF with large exchange interactions J/k B ∼ 250 K. 28 In the present work, we synthesized the new hexagonal compound Rb 2 Cu 3 SnF 12 with a similar crystal structure as Cs 2 Cu 3 ZrF 12 and performed magnetic susceptibility and high-field magnetization measurements using single crystals. As shown below, we found that the ground state is a disordered singlet with a finite gap for magnetic excitations. Rb 2 Cu 3 SnF 12 crystals were synthesized via the chemical reaction 2RbF + 3CuF 2 + SnF 4 → Rb 2 Cu 3 SnF 12 . RbF, CuF 2 , and SnF 4 were dehydrated by heating in vacuum at 60∼100 • C for three days. The materials were sealed in a platinum tube in the ratio of 3 : 3 : 2. Single crystals were grown from the melt. The temperature of the furnace was lowered from 800 to 550 • C for four days. Transparent colorless crystals with a typical size of 5×5×0.5 mm 3 were obtained.
Since the crystal structure of Rb 2 Cu 3 SnF 12 has not been reported to date, we performed a structural analysis at room temperature using a Bruker SMART-1000 threecircle diffractometer equipped with a CCD area detector. Monochromatic Mo-Kα radiation was used as an X-ray source. Data integration and global-cell refinements were performed using data in the range of 1.96 • < θ < 27.64 • , and multi-scan empirical absorption correction was also performed. The total number of reflections observed was 15268, and 1765 reflections were found to be independent and 1688 reflections were determined to satisfy the criterion I > 2σ(I). Structural parameters were refined by the full-matrix least-squares method using SHELXL-97 software. The crystal showed a merohedral twin structure. The twin law is (0, 1, 0 / 1, 0, 0 / 0, 0, −1) and the occupancy of the twin component was refined to 0.393(1). The final R indices obtained were R=0.020 and wR=0.056. The structure of Rb 2 Cu 3 SnF 12 is hexagonal, R3, with cell dimensions of a=13.917(2)Å and c=20.356(3)Å, and Z=12. Atomic coordinates, equivalent isotropic displacement parameters, and site occupancies are shown in Table I. Disorder was observed for the positions of F(6) and F(7). Site occupancies for F(6A) and F(7A) are 0.773 (8) and those for F(6B) and F(7B) are 0.227 (8). The crystal structure viewed along the c-axis is illustrated in Fig.  1(a), where F(4), F(5), F(6), and F(7) located outside the Kagomé layer are omitted, so that magnetic Cu 2+ ions and exchange pathways (gray bonds) are visible. The structure of Rb 2 Cu 3 SnF 12 is closely related to the Cs 2 Cu 3 ZrF 12 structure. 26 The chemical unit cell is described by enlarging that of Cs 2 Cu 3 ZrF 12 to 2a, 2a, c. CuF 6 octahedra are linked in the c-plane sharing corners. Magnetic Cu 2+ ions with S=1/2 located at the center of the octahedra form a Kagomé lattice in the c-plane. CuF 6 octahedra are elongated along the principal axes that are approximately parallel to the c-axis, so that the hole orbitals d(x 2 − y 2 ) of Cu 2+ spread in the Kagomé layer.  to 90 • . The exchange interactions J 1 ∼J 4 are labeled in decreasing order of α. Disorder in F(6) and F(7) may slightly affect the exchange interactions between Cu 2+ ions, because they are outside the Kagomé layer and do not take part in the exchange processes. The interlayer exchange interaction J ′ should be much smaller than J, because magnetic Cu 2+ layers are sufficiently separated by nonmagnetic Rb + , Sn 4+ , and F − layers. Thus, Rb 2 Cu 3 SnF 12 is expected to be a 2D S=1/2 HKAF. Magnetic susceptibilities of Rb 2 Cu 3 SnF 12 were measured in the temperature range 1.8−400 K using a SQUID magnetometer (Quantum Design MPMS XL). High-field magnetization measurement was performed using an induction method with a multilayer pulse magnet at the Institute for Solid State Physics, The University of Tokyo. Magnetic fields were applied parallel and perpendicular to the c-axis in both experiments. Figure 2 shows the temperature dependence of magnetic susceptibilities of Rb 2 Cu 3 SnF 12 measured at H=1 T. It is noted that one molar Rb 2 Cu 3 SnF 12 contains three molar Cu 2+ ions. The obtained susceptibility data were corrected for the diamagnetism χ dia of core electrons and for Van Vleck paramagnetism. The diamagnetic susceptibilities of individual ions were taken from the literature. 29 Van Vleck paramagnetic susceptibility was calculated using χ µ VV = −(N µ 2 B /λ)∆g µ = 3.14 ×  10 −4 ∆g µ emu/Cu 2+ mol, where λ = −829 cm −1 is the spin-orbit coupling coefficient of Cu 2+ and ∆g µ = g µ − 2 is the anisotropy of the g-factor. The g-factors used are g =2.44 and g ⊥ =2.15, which were determined selfconsistently by fitting the result of exact diagonalization to the experimental susceptibilities shown in Fig.  2. These g-factors are consistent with those parallel and perpendicular to the elongated axes of CuF 6 octahedra in K 2 CuF 4 and Rb 2 CuF 4 . 30 With decreasing temperature, the magnetic susceptibilities of Rb 2 Cu 3 SnF 12 exhibit rounded maxima at T max ∼70 K and decrease rapidly. Small upturn is observed below 7 K, which should be ascribed to impurities on crystal surfaces. No magnetic ordering is observed. This result indicates clearly that the ground state is a disordered singlet with a spin gap, as predicted from a recent theory on a 2D S=1/2 HKAF. Low-temperature susceptibilities are expressed as where the first term is the Curie-Weiss term due to impurities, the second term, the asymptotic low-temperature susceptibility for a 2D spin gap system, 31 and the last term, the constant term arising from the triplet component mixed in the ground state. Applying eq. (1) to the raw susceptibility data for T < 10 K, we evaluated the impurity contribution (≃0.3 %). Red symbols in Fig.  2 denote the intrinsic susceptibilities corrected for impurities. The susceptibility for H c is almost zero for T →0, whereas that for H⊥c is finite. In Rb 2 Cu 3 SnF 12 , elongated axes of CuF 6 octahedra incline alternately in the Kagomé layer, inducing a staggered field when an external field is applied. Since there is no inversion center in the middle of two neighboring magnetic ions in the Kagomé layer, the Dzyaloshinsky-Moriya (DM) interaction is allowed. The Zeeman interaction due to the staggered field and the DM interaction can have finite matrix elements between the singlet ground state and the excited triplet state, because they are antisymmetric with respect to the interchange of the interacting spins. Thus, we infer that the ground state has a small amount of triplet component through these antisymmetric interactions when subjected to the external field parallel to the Kagomé layer. This gives rise to the finite susceptibility at T =0. The magnitude of the spin gap is estimated to be ∆/k B = 21(1) K by fitting eq. (1) to the raw susceptibility data for H c. For T < 150 K, the magnetic susceptibility of the present system does not agree with the theoretical result for the S=1/2 uniform HKAF shown by red solid line in Fig. 3. 25 The theoretical susceptibility exhibits a sharp rounded maximum at T max ∼(1/6)J/k B , whereas the experimental susceptibility exhibits a broad maximum. This is because four nearest-neighbor exchange interactions are different. As shown in Fig. 1(b), one chemical unit cell in a Kagomé layer contains 12 spins. Then, we carried out the exact diagonalization for the 12-site Kagomé cluster under the periodic boundary condition. First, we calculated the uniform case, J 1 =J 2 =J 3 =J 4 =1, to check the validity of our calculation. The obtained result is shown by a black solid line in Fig. 3, where temperature and susceptibility are scaled using the average of exchange interactions J av = (1/4) i J i . The result is the same as that obtained by Elstner and Young, 8 and close to the result for the 24-site Kagomé cluster obtained by Misguich and Sindzingre. 25 Dashed and dotted lines in Fig. 3 denote the results for J 4 =1.5 and 0.5, respectively, where we set J 1 =J 2 =J 3 =1. Whether for J 4 > 1 or J 4 < 1, the maximum susceptibility χ max decreases and T max shifts toward the high-temperature side, as observed in the present measurements. We also investigated the effects of J 2 and J 3 , setting the others unity. For J 2 < 1, χ max increases and T max decreases with decreasing J 2 , whereas for J 2 > 1, the susceptibility diverges for T →0. This is because six spins coupled by J 2 on a hexagon form a singlet state and three spins coupled by J 3 on a triangle form a doublet state. The susceptibility is not largely affected by J 3 .
The antiferromagnetic exchange interaction through F − ion becomes stronger with increasing bonding angle α of the exchange pathway Cu 2+ −F − −Cu 2+ . Since α 1 =138.4 • , α 2 =136.4 • , α 3 =133.4 • , and α 4 =123.9 • , the condition J 1 > J 2 > J 3 > J 4 must be realized in the present system. Under this condition, we calculated susceptibility. The best fit for T > T max was obtained using J 1 /k B =234(5) K, J 2 /k B =211(5) K, J 3 /k B =187(5) K, and J 4 /k B =108(5) K with g =2.44(1) and g ⊥ =2.15 (1). Solid lines in Fig. 2 indicate the susceptibilities calculated with these parameters. These exchange parameters are valid from the fact that J/k B =244 K and α=141.6 • in Cs 2 Cu 3 ZrF 12 , 28 and J/k B =103 K and α=129.1 • in KCuGaF 6 . 32 For T < T max , the calculated susceptibility decreases more rapidly than the experimental susceptibility. This should be ascribed to the finite-size effect. Calculation for a larger cluster may give a better description of low-temperature susceptibility.
To evaluate the spin gap directly, we performed highfield magnetization measurements. Figure 4 shows magnetization curves measured at T =1.3 K for H c and H⊥c. The magnetization is small up to the critical field H c indicated by arrows and increases rapidly. The levels of the ground and excited states cross at H c . The magnetization does not show a sharp bend at H c , but is rather rounded. We infer that the antisymmetric interactions, such as the staggered Zeeman and DM interactions, give rise to the smearing of the magnetization anomaly. We assign the critical field H c to the field of inflection in dM/dH. The critical fields obtained for H c and H⊥c are H c =13(1) T and 20(1) T, respectively. These critical fields do not agree when normalized by the g-factor as (g/2)H c . When an external field is applied perpendicular to the c-axis, the magnetic susceptibility is finite even at T =0. Consequently, the ground state energy is not independent of the external field but decreases with the external field, resulting in an increase in the critical field. Therefore, as the spin gap, we take ∆/k B =21(1) K obtained from H c =13(1) T for H c. This spin gap is the same as that evaluated from low-temperature susceptibility and is approximately one-tenth of J av .
Since the largest interaction is the J 1 interaction, the ground state is composed of singlet dimers on the J 1 bonds. Although the J 2 and J 3 interactions are smaller than the J 1 interaction, their magnitudes are similar. Thus, the local change of dimer positions does not cost so large amount of energy as to create a local triplet state. Such singlet excitations may propagate in a crystal as singlet wave. We infer that there are many singlet states inside the spin gap, as predicted for the proper HKAF, which should be verified by specific heat measurement.
In conclusion, we have presented the results of structural analysis and magnetic measurements of Rb 2 Cu 3 SnF 12 . The crystal structure is hexagonal and closely related to the Cs 2 Cu 3 ZrF 12 structure with proper Kagomé layers of Cu 2+ . Rb 2 Cu 3 SnF 12 can be described as a 2D S=1/2 modified HKAF with four kinds of neighboring exchange interaction. The results of magnetic susceptibility and high-field magnetization measurements revealed that the ground state is a disordered singlet with a spin gap, as predicted from a recent theory. We performed exact diagonalization for a 12-site Kagomé cluster to analyze the magnetic susceptibility and evaluated individual exchange interactions.