Abstract
Molecular quantum magnetism involving an isolated spin state is of particular interest due to the characteristic quantum phenomena underlying spin qubits or molecular spintronics for quantum information devices, as demonstrated in magnetic metal–organic molecular systems, the socalled molecular magnets. Here we report the molecular quantum magnetism realized in an inorganic solid Ba_{3}Yb_{2}Zn_{5}O_{11} with spin–orbit coupled pseudospin½ Yb^{3+} ions. The magnetization represents the magnetic quantum values of an isolated Yb_{4} tetrahedron with a total (pseudo)spin 0, 1 and 2. Inelastic neutron scattering results reveal that a large Dzyaloshinsky–Moriya interaction originating from strong spin–orbit coupling of Yb 4f is a key ingredient to explain magnetic excitations of the molecular magnet states. The Dzyaloshinsky–Moriya interaction allows a nonadiabatic quantum transition between avoided crossing energy levels, and also results in unexpected magnetic behaviours in conventional molecular magnets.
Introduction
Quantum magnetism is a ubiquitous subject from the spin singlet state in noninteracting dimers to the longrange entangled state in quantum spin liquids^{1,2,3,4,5,6,7,8}. Its nature is often described in terms of quantum values of the magnetic moments, ionic anisotropies and their coupling network^{9,10,11,12,13}. The former two are determined by the single ionic characters, while the last one can be controlled by arrangement of the magnetic ions. Great progress in theoretical and experimental investigations on the quantum magnetism has mostly been made through variation of the coupling connectivity^{3,4,5,6,7,8,9,10,11,12,13}. An extreme case could be the molecular magnet of an isolated magnetic cluster^{14,15}. Meanwhile, magnetism in solids is strongly influenced by geometrical constraints such as frustration or bond alternation as well as dimensionality, and often introduces intriguing quantum phenomena^{1,5,6,7,8,16}.
With strong spin–orbit coupling (SOC), the magnetic quantum spin is given by the total angular momentum J, rather than the spin S. The degenerate Jstate is split by a crystal field (CF), and the groundstate quantum spin can be represented by a modified J, the socalled pseudospin, as discussed in lanthanidebased molecular magnets^{17,18,19,20,21,22}. The strong 4f SOC combined with a subtle CF contributes strong magnetic anisotropy to yield a simple model Hamiltonian with Isinglike or XYlike anisotropic magnetic exchange between the molecular pseudospins. Recently, Ba_{3}Yb_{2}Zn_{5}O_{11} was reported to be a geometrically frustrated breathing pyrochlore system with two distinct Yb–Yb distances^{23,24}. Ba_{3}Yb_{2}Zn_{5}O_{11} consists of two alternating main blocks of Yb_{4}O_{16} and Zn_{10}O_{20} (Fig. 1a). The magnetic Yb^{3+} ions in Yb_{4}O_{16} form a tetrahedron connected with another one through cornersharing in the threedimensional framework. Remarkably, the intertetrahedron Yb–Yb distance r′=6.23 Å is much larger than the intratetrahedron one r=3.30 Å. Thus, the intermagnetic exchange energy J′ becomes negligible in comparison with the intraexchange energy J, that is, J′/J∼0 (Fig. 1b). As can be seen in the detailed ionic arrangements of the tetrahedron Yb_{4}O_{16} (Fig. 1c), each Yb ion is surrounded by six oxygens to form an octahedron (YbO_{6}) with a trigonal distortion (C_{3v} symmetry), and the C_{3v} symmetry axis is towards the tetrahedron center. The Yb^{3+} ion (4f^{13}) effectively has a pseudospin½ ground state of a Kramers doublet separated by the CF splitting energy of 38.2 meV (=443 Kk_{B}) without any magnetic longrange order even below subKelvin in spite of the considerable Curie–Weiss temperature Θ_{CW}=−6.7 K (refs 24, 25), indicating possible formation of decoupled molecular spin states.
In this paper, we report the molecular magnetic behaviours in magnetization and inelastic neutron scattering (INS) for inorganic polycrystalline Ba_{3}Yb_{2}Zn_{5}O_{11} samples. The magnetization shows the hysteretic steplike jumps between S_{eff}=0, 1 and 2 molecular magnetic states of an isolated Yb_{4} tetrahedron with spin–orbit coupled pseudospin½, which reflects the nonadiabatic Landau–Zener transition. The INS measurement with external magnetic field unveils that the large Dzyaloshinsky–Moriya (DM) interaction originating from strong SOC of Yb 4f electron is essential to construct the S_{eff} molecular magnetic states involving avoided level crossing. Our finding not only opens a possibility for qubit quantum device applications due to benefit of the regularity in the inorganic solid, but the tunable intertetrahedron distance also provides a playground to explore the crossover from isolated to entangled magnetic quantum systems in the presence of SOC.
Results
Magnetization and effective Hamiltonian
Figure 1d shows the fielddependent magnetization result (M versus H) at 100 mK, which displays the characteristic steplike jumps of an antiferromagnetic (AFM) coupled molecule with a total (pseudo)spin S_{eff} formed by the four ½pseudospins of the tetrahedron. The ground state of the AFMcoupled tetrahedron, which is the S_{eff}=0 state at zero field, is consecutively switched to the S_{eff}=1 and the S_{eff}=2 state as the external magnetic field increases across corresponding critical fields. The three plateaus in M(H) represent the respective S_{eff}=0, 1 and 2 states, and the consecutive level crossing quantum transitions are presented by the twostep feature at the critical fields H_{C1}=3.5 T and H_{C2}=8.8 T. It is worth to note that M(H) exhibits three distinctive features: a hysteretic behaviour near H_{C1}, shifts of the critical fields with respect to the Heisenberg model and a nonzero slope in a lower field region 0<H<H_{C1}, all of which are not expected in conventional molecular magnets with a simple Heisenberg AFM exchange interaction. For comparison, the simulated M(H) of the conventional molecular magnet with AFMcoupled four ½spins, which has H_{C1}=3.7 T and H_{C2}=7.4 T with no slope in 0<H<H_{C1}, is presented by the black dashed line in the figure.
Considering the pseudospin½ with the Yb 4f strong SOC, we should adopt a generalized magnetic exchange Hamiltonian^{17,18,19,26,27,28} to explain the observed magnetization;
The first and second terms represent the generalized magnetic exchange and Zeeman terms, respectively. The ith site pseudospin can be represented by the Pauli spin operator S_{i}, and the exchange coupling J_{ij} between S_{i} and S_{i} is presented in a tensor form (J^{μv}). The Zeeman term is described with the Bohr magneton μ_{B}, external magnetic field H and a gtensor g_{i} reflecting the ith site singleion magnetic anisotropy. The generalized Hamiltonian well explains observed INS results of the breathing pyrochlore Ba_{3}Yb_{2}Zn_{5}O_{11} with only a few nonetrivial J^{μv} values as discussed below, and is effectively reduced to an effective exchange Hamiltonian with an additional DM interaction (antisymmetric exchange interaction) term to the conventional Heisenberg AFM Hamiltonian (Supplementary Note 1):
The first term represents the Heisenberg exchange interaction and the second term accounts for the DM interaction, which is a result of combined effects of the SOC and the superexchange interactions. According to the Moriya’s rules^{28,29}, the DM vector d_{ij} does not disappear in Ba_{3}Yb_{2}Zn_{5}O_{11} (F3m space group) with noinversion symmetry, and its direction (green arrows in the inset of Fig. 1c) is constrained by tetrahedron symmetry (T_{d}).
The steplike features in M(H) at H_{C1}=3.5 T and H_{C2}=8.8 T reflect the spin singlet–triplet (S_{eff}=0–1) and triplet–quintet (S_{eff}=1–2) level crossings, respectively. This M(H) curve can be well reproduced with the respective gfactor values g_{}=3.0(1) and g_{∥}=2.4(1) for parallel and perpendicular to the symmetry axis of each YbO_{6}, respectively (green line in Fig. 1d). These values are slightly larger than g_{}=2.54 and g_{∥}=2.13, which are more accurately determined from an electron paramagnetic resonance measurement (Supplementary Note 2). It is noticed that a hysteretic behaviour appears around H_{C1}. This behaviour reflects the Landau–Zener transition involving an avoided level crossing with an energy gap in a dissipative twostate model (inset of Fig. 1d)^{30,31,32,33}. The level crossing energy gap hinders the adiabatic crossover between the singlet and triplet as the magnetic field increases or decreases across H_{C1}. Evidently, we observed that the hysteresis varies with the field sweep rate (Supplementary Note 3). At H_{C2}, the energy gap is enhanced, and the hysteretic feature becomes less effective. A finite slope in M(H) can be also noticed in the range 0<H<H_{C1}. Due to the fact that the DM interaction admixes the singlet and triplet states, the ground state is no longer a pure S_{eff}=0 state and the admixed triplet S_{eff}=1 state contributes the weak field dependence to the magnetization^{34}. While the anisotropic Zeeman term also can contribute to paramagnetic response to the applied field, its effect on M(H) is less significant than that of DM in the breathing pyrochlore Ba_{3}Yb_{2}Zn_{5}O_{11} (Supplementary Note 3).
INS without a magnetic field
To explore magnetic excitations in this novel quantum magnet, we performed INS measurements^{20,21,22,35,36}. Figure 2a shows the intensities I(Q,ω) as a function of momentum and energy transfer obtained from the measurements at T=200 mK and at the zero magnetic field. The I(Q, ω) exhibits four nondispersive excitations, which correspond to the transitions from the S_{eff}=0 ground state to the S_{eff}=1 excited states. As temperature increases, the S_{eff}=1 states become partially occupied due to the thermal energy, and additional transitions from the S_{eff}=1 states become available in the INS result. Indeed, we could observe additional nondispersive excitations at 10 K as shown in Fig. 2b.
To confirm the molecular characteristics of the quantum spin state, we examined the Qdependences of two dominant excitation intensities I(Q), which are integrated over 0.45–0.6 meV and 0.65–0.8 meV regions in Fig. 2a. The obtained I(Q) is compared with the calculated ones for an Yb^{3+} single ion (black dashed line) and an Yb_{4} tetrahedron molecule (green dashed line) as shown in Fig. 2c. The obtained I(Q) is obviously well explained by the molecular model rather than the ionic model^{35}, supporting the presence of molecular quantum magnetism in Ba_{3}Yb_{2}Zn_{5}O_{11}. The Q integrated intensities I(ω) presented in Fig. 2d are also well understood in a framework of the S_{eff} molecular magnet states. At 200 mK, we identified four excitation peaks, while seven peaks are observable at 10 K, which could be indexed with 11 excitations. The corresponding excitations (vertical arrows) are described in the energy level diagram based on the effective Hamiltonian , which is schematically depicted in Fig. 2e.
The Heisenberg exchange splits total 2^{4} magnetic states of the Yb_{4} molecule, consisting of four ½ pseudospins, into doubly degenerated S_{eff}=0, triply degenerated S_{eff}=1 and nondegenerated S_{eff}=2 states. Due to the DM interaction, the 3 × 3 S_{eff}=1 states are admixed with the lowest S_{eff}=0 states and split into four states, ψ_{1}(1), ψ_{2}(3), ψ_{3}(3) and ψ_{4}(2), with degeneracies presented in the parenthesis, while the ground state ψ_{0}(2) is further lowered in energy. On the other hand, the S_{eff}=2 quintet state ψ_{5}(5) is not affected. For detailed analyses, we calculated eigenstates and eigenvalues of using the exact diagonalization method, and estimated the magnetic scattering crosssections. With optimized values of J=0.589 meV and d=0.158 meV (refs 37, 38), we obtained simulated I(ω) spectra at 200 mK (blue line) and 10 K (red line), which well reproduce the experimental data as shown in Fig. 2d. The obtained large DM value (d/J=0.27), which reflects the strong SOC of Yb^{3+} ions, is consistent with the value theoretically estimated for the Yb–O–Yb superexchange hopping (Supplementary Note 4).
INS with magnetic fields
Validity of the effective Hamiltonian can be also confirmed in the fielddependent INS results at 200 mK. Figure 3a shows I(ω), integration of I(Q,ω) over 0.8 Å^{−1}<Q<1.8 Å^{−1}, under various external magnetic fields in comparison with the theoretical simulations. The excitation peaks evolve with the external field. The simulated I(ω) spectra represent spherically averaged I(ω) from exactly diagonalized . The simulations well reproduce the experiments with the gfactor values g_{}=2.62(2) and g_{∥}=2.33(2), which are slightly smaller than the values obtained from the M(H) curve, likely due to an estimation error. These gfactor values are also consistent with the values estimated from the electron paramagnetic resonance spectrum and those determined from YbO_{6} CF analyses for reported highenergy neutron excitation spectra^{25} as discussed in Supplementary Notes 2 and 4, respectively.
Figure 3b shows the calculated excitation energy diagrams as a function of external magnetic field H along the principle axes [0 0 1], [1 1 0] and [1 1 1] using the obtained values of the J, d and gfactors. The colour of the excitation energy line represents the magnetization value <g S^{z}> ranged from −g_{}/2 to +g_{}/2 as presented by a colour scale bar in Fig. 3c. One can notice that the overall energy diagram is almost identical for the three axes except minor variations in the excitation energies, which appear as peak broadenings in the observed I(ω) in Fig. 3a, and the calculated energies coincide with the peak positions. We also trace three excitation peaks as marked in Fig. 3a,b. The first excitation (diamond) corresponds to ψ_{0} (S_{eff}≈0)→ψ_{1} (S_{eff}≈1) for H<H_{C1} (≈3.5 T). For H>H_{C1}, the Zeeman energy of ψ_{1} overcomes their zerofield energy difference, that is, level crossing, and the excitation is switched to the second one (star), ψ_{1}→ψ_{0}. As H exceeds H_{C2}, ψ_{5} (S_{eff}=2) becomes the ground state, and the third one (triangle) representing the excitation ψ_{1}→ψ_{5} becomes unavailable. The avoided level crossing feature around H=H_{C1} is also examined in the calculated excitation. Figure 3c shows the ψ_{0}→ψ_{1} to ψ_{1}→ψ_{0} excitation crossover for H//[1 1 0] in a verylowenergy region. Even though H approaches H_{C1}, the excitation energy between ψ_{0} and ψ_{1} does not vanish, confirming existence of a finite gap, that is, avoided level crossing, which originates the hysteretic behaviour of M(H) discussed above. The estimated gap energy Δ≈0.013 meV corresponds to a gigahertz range in a qubit model system.
Discussion
Our analysis demonstrates that the inorganic solid Ba_{3}Yb_{2}Zn_{5}O_{11} realizes a novel molecular quantum magnet. By virtue of strong SOC of Yb 4f electrons, each Yb^{3+} ion has a spin–orbit coupled pseudospin½, and the antisymmetric DM exchange interaction plays a crucial role in the magnetism. This quantum magnetism exhibits not only the exotic Landau–Zener transition involving avoided level crossing but also paramagnetic responses under a magnetic field that differ significantly from conventional molecular magnetism. In this inorganic material, the intermolecular exchange coupling J′ is negligible, but as the intermolecular spacing is reduced by the substitution of nonmagnetic ions with smaller ionic sizes or by applying pressure, J′ increases to turn on the entanglement between the molecular spins. A weakly coupled alternating pyrochlore system can be considered as a protocol for quantum gates and spin manipulations by electric fields, as proposed for weakly coupled molecular spin triangles^{39}. When J′ becomes significant, the system ends to be a typical frustrated magnet^{23}. Therefore, this material provides a promising starting point for exploration of an uncharted crossover from molecular to entangled quantum magnetism.
Methods
Sample synthesis and magnetization
Polycrystalline Ba_{3}Yb_{2}Zn_{5}O_{11} samples were prepared by the solidstate reaction method from a stoichiometric mixture of Ba_{2}CO_{3} (99.999%), Yb_{2}O_{3} (99.99%) and ZnO (99.999%) powders as in the (ref. 24). The mixture pellet was successively sintered at 1,000 °C for 24 h and at 1,120 °C for 24 h in air. The isothermal magnetization was determined for a 60 mg of pelletized sample below T=1.8 K using a conventional Faraday force magnetometer, which measures changes in the electric capacitance induced by a magnetic field gradient. To avoid any movement of grains during the measurements, we pelletized a 60 mg of powder to a cubicshaped hard solid, and firmly mounted it on the magnetometer sample holder, and no crack was observable on the pellet after the measurements. The measurements were performed under a static magnetic field in a range from 0 to 12 T with various sweep rates from 7.5 to 30 mT min^{−1}. The magnetization at 1.8 K, which was measured using the Quantum Design’s Magnetic Property Measurement System, was utilized as a reference to determine the magnetizations at different temperatures.
Elastic and inelastic neutron scattering
Neutron powder diffraction experiment was conducted at the highresolution powder diffraction beamline in HANARO for the structural information (Supplementary Fig. 1; Supplementary Table 1). Powder samples of 5 g were sealed in vanadium container and placed on a closed cycle refrigerator for the diffraction measurements at 4 K. The crystal structure was determined from the Rietveld refinement using the FullProf suite software^{40}. Timeofflight neutron scattering experiment was carried out at the Disk Chopper Spectrometer beamline in the National Institute of Standards and Technology Center for Neutron Research. The neutron scattering data were obtained at T=200 mK and 10 K. The incident neutron energy was set to be 2.27 meV (=6 Å) with an energy resolution of 64 μeV at the elastic line. A polycrystalline sample of 10 g was sealed in a copper container, and inserted into a 10 T vertical field magnet equipped with a dilution refrigerator.
The inelastic magnetic neutron scattering intensity for isolated Yb_{4} tetrahedrons is given by
where I_{0} is a scale factor, k_{i} and k_{f} are initial and final neutron wave vectors, and Q and ω are the momentum and energy transfers, respectively. F(Q) is a dimensionless magnetic form factor of Yb^{3+}, and α, β=x, y and z, and the dynamical structure factor is defined as
where i is the site index and is an eigenstate of with an energy . denotes the thermal population with the partition function Z, and M_{i} is the isite effective magnetic moment . For a powder sample with an external magnetic field H, the inelastic magnetic neutron scattering intensity is obtained by averaging I(Q, ω) over all directions of Q^{41} and H as
which is adopted for the INS data fitting.
Code availability
We declare that the datasimulation code supporting the findings of this study are available within article’s Supplementary Information file (Supplementary Software 1).
Data availability
The data that support the findings of this study are available from the corresponding author on request.
Additional information
How to cite this article: Park, S.Y. et al. Spin–orbit coupled molecular quantum magnetism realized in inorganic solid. Nat. Commun. 7, 12912 doi: 10.1038/ncomms12912 (2016).
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Acknowledgements
We became aware of the experimental works by Rau et al.^{37} and Haku et al.^{38} after we have submitted this paper. They also obtained the consistent results with ours using INS without an external magnetic field and the same model Hamiltonian. In our work, we have advanced by showing that the DM interaction associated with the avoided energy level crossing plays a crucial role for unconventional quantum magnetic behaviours such as a hysteresis and paramagnetic response in magnetization, which is confirmed by INS with an external magnetic field. We are grateful to K.B. Lee and K.S. Park for enlightning discussions. This work is supported by the National Research Foundation (NRF) through the Ministry of Science, ICP & Future Planning (MSIP) (No. 2016K1A4A4A01922028). B.H.K. is supported by the RIKEN iTHES Project. S.H.D. and K.Y.C. are supported by the Korea Research Foundation grant (No. 20090076079) funded by the Korea government (MEST). S.L. is supported by the NRF under the contract NRF2012M2A2A6002461.
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Contributions
S.Y.P., N.P.B. and S.J. performed INS measurement. J.H.K., D.J. and M.B. carried out magnetization measurement. S.Y.P., S.L. and S.J. performed neutron powder diffraction measurement. S.H.D. and K.Y.C. synthesized samples. S.Y.P., B.H.K., B.S. and S.J. developed the theoretical model of the effective Hamiltonian. D.H.K., B.H.K. and J.H.P. performed CF and superexchange model calculations. S.Y.P. and S.J. analysed the data. S.Y.P., S.J. and J.H.P. wrote the manuscript. J.H.P. and S.J. initiated and supervised the research.
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Correspondence to J.H. Park or Sungdae Ji.
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Supplementary information
Supplementary Information
Supplementary Figures 16, Supplementary Table 1, Supplementary Notes 14 and Supplementary References. (PDF 1242 kb)
Supplementary Software
Mathematica notebook which calculates eigenstates of a four1/2(pseudo)spin tetrahedron by full diagonalization of the effective Hamiltonian and simulates a magnetisation, a magnetic susceptibility and a specific heat without a directional average. (TXT 318 kb)
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Park, S., Do, S., Choi, K. et al. Spin–orbit coupled molecular quantum magnetism realized in inorganic solid. Nat Commun 7, 12912 (2016). https://doi.org/10.1038/ncomms12912
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