Abstract
Geometric frustration, in which competing interactions give rise to degenerate ground states, potentially induces various exotic quantum phenomena in magnetic materials. Minimal models comprising triangular units, such as triangular and Kagome lattices, have been investigated for decades to realize novel quantum phases, such as quantum spin liquid. A pentagon is the secondminimal elementary unit for geometric frustration. The realization of such systems is expected to provide a distinct platform for studying frustrated magnetism. Here, we present a spin1/2 quantum pentagonal lattice in the new organic radical crystal α2,6Cl_{2}V [=α3(2,6dichlorophenyl)1,5diphenylverdazyl]. Its unique molecular arrangement allows the formation of a partially cornershared pentagonal lattice (PCPL). We find a clear 1/3 magnetization plateau and an anomalous change in magnetization in the vicinity of the saturation field, which originate from frustrated interactions in the PCPL.
Introduction
Closed loop lattice systems with an odd number of antiferromagnetic (AFM) bonds induce frustration through competing exchange interactions that cannot be simultaneously satisfied. Pentagonal lattices can therefore induce frustration in analogy with systems based on triangular lattice, which have been investigated extensively^{1,2,3,4,5,6,7}. Quantum pentagonal systems have yet to be realized experimetaly however. Regular pentagons cannot tile a plane because of the crystallographic restriction theorem, such that distortion and/or additional shapes are necessary^{8,9,10,11}. The Cairo pentagonal lattice—a twodimensional plane consisting of distorted pentagons with two inequivalent sites^{8}—has attracted considerable attention since its recent realization in ironbased compounds with classical spins^{12,13}. Although the lattice systems in ironbased compounds differ somewhat from the regular Cairo pentagonal lattice, their realizations have inspired further theoretical studies on quantum cases^{14,15,16,17}, where the emergence of a spinnematic phase and a 1/3 magnetization plateau are predicted. These specific quantum phases originate from the two types of inequivalent sites and the six spins in the magnetic unit cell, important characteristics that are common to the Cairo pentagonal lattice and the partially cornershared pentagonal lattice (PCPL) investigated here.
Here, we introduce the basic properties of the PCPL. It contains two inequivalent sites, α and β, with coordination numbers 2 and 4, respectively, as shown in Fig. 1(a). The α site has two β neighbors, whereas each β site is connected to one α and three β sites. One ferromagnetic (FM) interaction J_{1} and two AFM interactions J_{2} and J_{3} form a twisted pentagonal unit consisting of J_{1}J_{3}J_{2}J_{3}J_{1} and induce frustration. The unit cell of the PCPL contains two α and four β sites (see Supplementary Information). The six spins in the unit cell and the observed 1/3 magnetization plateau suggest the existence of a nonmagnetic singlet state with an excitation energy gap formed by the AFM interactions J_{2} and/or J_{3} between the β sites. The residual αsite spins interact with one another through the triplet excited states of such singlet state.
The symmetry and shape of electron orbitals make crystals based on pentagonal lattices difficult to form in inorganic materials. In fact, there are few examples in the history of condensed matter physics. Unconventional lattice system should however be realizable with organic radical materials in diverse molecular arrangements. We recently established synthetic techniques for the preparation of highquality verdazyl radical crystals^{18}. In contrast to other conventional radicals such as nitroxide and nitronyl nitroxide, the πelectron spin density of verdazyl radicals can be delocalized even in nonplanar molecular structures. This makes the molecular orbitals (MOs) associated with exchange interactions flexible in shape and enables the design of lattice systems by chemical modification, thereby facilitating the synthesis of new materials forming unconventional lattice systems^{18,19,20,21}. Here, we present the successful synthesis of a new verdazyl radical crystal, α2,6Cl_{2}V. Ab initio MO calculations indicate the formation of an S = 1/2 quantum pentagonal lattice consisting of one FM and two AFM interactions.
Results and Discussion
Figure 1(b) shows the molecular structure of 2,6Cl_{2}V. The crystallographic parameters at room temperature are as follows^{22}: orthorhombic, space group Fdd2, a = 42.759(3) Å, b = 15.5551(12) Å, c = 16.6127(13) Å, V = 11049.4(15) Å^{3}, Z = 24, R = 0.0346 and R_{w} = 0.0846. Furthermore, there is no indication of a structural phase transition down to around 23 K (see Supplementary Table S1). The central verdazyl ring with four nitrogen atoms and three phenyl rings are labeled R_{1}, R_{2}, R_{3} and R_{4}, respectively [Fig. 1(b)]. The crystals contain two crystallographically independent molecules, in which the large ionic radius of the Cl atom, introduced at the 2,6position, induces a relatively large dihedral angle (>80°) at R_{1}R_{3} owing to electrostatic repulsion between the Cl and N atoms [Fig. 1(b)]. This nonplanar structure inhibits molecular stacking and the overlap of singly occupied MOs, resulting in the formation of the PCPL.
Ab initio MO calculations on the basis of the crystallographic arrangement at 23 K show that one FM and two AFM exchange interactions, labeled as J_{1}, J_{2} and J_{3} (see Supplementary Fig. S2), are dominant. They are evaluated as J_{1}/k_{B} = −1.6 K, J_{2}/k_{B} = 3.9 K and J_{3}/k_{B} = 3.3 K, which are defined in the Heisenberg spin Hamiltonian given by , where denotes the sum over the neighboring spin pairs. The three evaluated interactions form a twisted pentagonal unit consisting of J_{1}J_{3}J_{2}J_{3}J_{1}, as shown in Fig. 1(c). The pentagonal units are connected to one another by sharing a corner, resulting in the formation of the S = 1/2 PCPL, as shown in Fig. 1(a).
Figure 2(a) shows the temperature dependence of the magnetic susceptibilities (χ = M/H) at 0.1 T. At temperature above 50 K, the CurieWeiss law is followed, χ = C/(T − θ_{W}). The estimated Curie constant is about C = 0.362 emu· K/mol, which is close to the expected value for noninteracting S = 1/2 spins and the Weiss temperature is estimated to be θ_{W} = −0.85(5) K. Considering the meanfield approximation expressed as , this small absolute value of θ_{W} indicates a weak internal field due to competition between the FM and AFM interactions. We observe a shoulder and corresponding twostep decrease in χΤ at about 1 K [inset of Fig. 2(a)] , which indicate the contribution of two or more types of AFM interactions.
Figure 2(b) shows the temperature dependence of the total specific heat, C_{p}, at zerofield. A clear broad peak appears at about 1.4 K, which is characteristic of a Schottkylike behavior associated with an energy gap between excited states. Consistent with such Schottkylike C_{p} behavior, a clear 1/3 magnetization plateau is observed from 0.4 to 1.3 T, indicating an energy gap between the excited states, as shown in Fig. 4. The six spins in the unit cell and the 1/3 magnetization plateau suggest the existence of a nonmagnetic singlet state with an excitation energy gap formed by the AFM interactions J_{2} and/or J_{3} between the β sites. If such is the case, residual αsite spins interact with one another through the excited states of the singlet state as discussed later. The magnetization curve indeed increases less steeply than the Brillouin function for free S = 1/2 spins up to the plateau phase, highlighting the presence of nonnegligible AFM internal fields. In this context, a sharp peak of the specific heat observed at about 0.07 K is explained by a zerofield phase transition to longrange AFM order of the αsite spins. In the 1/3 plateau phase, these spins are fully polarized along the external field direction such that the ordered state disappears. The phase transition actually disappears in the specific heat at 1.0 T, while the Schottkylike behavior is shifted to around 0.5 K in response to the decrease of the energy gap, as shown in Fig. 3. In the case of the higher field region above the plateau phase, the field derivative of the magnetization curve (dM/dH) indicates linear behavior in H near the saturation field from 3.3 to 4.0 T, as shown in Fig. 4. This behavior is in clear contrast to the magnetization curve of conventional quantum spin systems, in which external field suppresses quantum fluctuations, which results in an upward curvature of the magnetization curve and a sharp dH/dH peak at the phase transition to fully polarized state^{23,24}. The upturn observed in C_{p} at 4.0 T should therefore be attributed to unconventional magnetic behavior near the saturation field.
In order to identify the contribution of abovementioned AFM interaction, we assumed a simple situation in which the fully polarized αsite spin works only as an internal field for each βsite spin in the highfield region above the 1/3 magnetization plateau. Accordingly, we considered a lattice system consisting of only J_{2} and J_{3} with an effective internal field , where H_{ext} is the external magnetic field. In the extreme case where , the two spins connected by the J_{2} interaction form a nonmagnetic singlet dimer with an excitation energy gap. In gapped cases such as this, the magnetization curve near the critical field—the end of the 1/3 plateau phase in the present case—increases with the square root of the applied field. For in contrast, the AFM interaction J_{3} forms a wellknown S = 1/2 uniform AFM chain with a TomonagaLuttinger liquid ground state and the energy gap disappears. The 1/3 plateau phase and sharp dM/dH peak that are observed here therefore indicate a singlet state arising from the AFM interaction J_{2} with , as described in Fig. 4. Accordingly, the effective interactions between the residual αsite spins are caused through the triplet excited states of the J_{2} singlet dimer^{22,25,26}. They are roughly evaluated from the secondorder and thirdorder perturbation treatment of the J_{2} term in the spin Hamiltonian (see Supplementary Information). We can consider that αsite spins form a uniform AFM chain with weak interchain interactions in the lowfield region. These interactions should cause the phase transition to longrange AFM order of the αsite spins.
Finally, concerning the unconventional behavior near the saturation field, a plausible explanation is that this reflects a hidden order of spin multipoles caused by correlations between multimagnon bound states, such as those in a spinnematic phase^{27}. The presence of such a phase is expected in the highfield region (near the saturation field) in S = 1/2 frustrated spin systems with FM interactions^{27,28}, but has not been verified experimentally to date. In the PCPL, the three spins connected by the J_{1} FM interaction may stabilize the threemagnon bound state in the vicinity of the saturation field, resulting in multipole spin order. Additional experimental techniques, notably neutron scattering using deuterated samples, should afford a more quantitative description of this fieldinduced phase and clarify the pentagonal frustration effect.
Methods
We synthesized 2,6Cl_{2}V using a conventional procedure similar to that used for preparing the typical verdazyl radical 1,3,5triphenylverdazyl^{29}. The crystal structure was determined on the basis of intensity data collected using a Rigaku AFC8R Mercury CCD RAMicro7 diffractometer with Japan Thermal Engineering XRHR10K. The magnetizations were measured using a commercial SQUID magnetometer (MPMSXL, Quantum Design) and a capacitive Faraday magnetometer down to about 70 mK. The experimental results were corrected for diamagnetic contribution (−2.57 × 10^{−4}), which is determined to become almost χΤ = const. above about 200 K and close to the value calculated by Pascal's method. The specific heat was measured using a handmade apparatus by a standard adiabatic heatpulse method down to about 50 mK. Considering the isotropic nature of organic radical systems, all experiments were performed using small randomly oriented single crystals. The ab initio MO calculations were performed using the UB3LYP method as brokensymmetry hybriddensity functional theory calculations. All the calculations were performed using the Gaussian 09 program package and the basis functions used were 6–31G. To estimate the intermolecular magnetic interaction of the molecular pairs within 4.0 Å, we applied our previously presented evaluation scheme^{30}.
Additional Information
How to cite this article: Yamaguchi, H. et al. Experimental Realization of a Quantum Pentagonal Lattice. Sci. Rep. 5, 15327; doi: 10.1038/srep15327 (2015).
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Acknowledgements
We thank T. Kawakami, H. Nakano and T. Tonegawa for valuable discussions. We also thank A. Matsuo for specific heat measurement. This research was partly supported by a Grant for Basic Science Research Projects from the Sumitomo Foundation, KAKENHI (Nos. 24540347 and 24340075), the Strategic Programs for Innovative Research (SPIRE), MEXT and the Computational Materials Science Initiative (CMSI), Japan. This work was partly performed under the interuniversity cooperative research program of the jointresearch program of the ISSP (University of Tokyo) and the Institute for Molecular Science.
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H.Y., K.I., N.A., T.O. and Y.H. carried out sample preparation and characterization. H.Y. and T.O. discussed the results. S.K., T.S. and K.A. performed the magnetic and thermodynamic measurements.
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Yamaguchi, H., Okubo, T., Kittaka, S. et al. Experimental Realization of a Quantum Pentagonal Lattice. Sci Rep 5, 15327 (2015). https://doi.org/10.1038/srep15327
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