Spin–orbit coupled molecular quantum magnetism realized in inorganic solid

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 so-called molecular magnets. Here we report the molecular quantum magnetism realized in an inorganic solid Ba3Yb2Zn5O11 with spin–orbit coupled pseudospin-½ Yb3+ ions. The magnetization represents the magnetic quantum values of an isolated Yb4 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 non-adiabatic quantum transition between avoided crossing energy levels, and also results in unexpected magnetic behaviours in conventional molecular magnets.


Supplementary Note 1. Effective Hamiltonian and inelastic neutron scattering
The Yb ion in Ba3Yb2Zn5O11 is surrounded by six oxygens, i.e. YbO6 octahedron, and four neighboring Yb ions form an isolated tetrahedron. The octahedron is slightly distorted in a space group F4 ̅ 3m . Each Yb site in the isolated tetrahedron is under a 3v local symmetry and the threefold axes are located along the [1 -1 -1], [-1 1 -1], [-1 -1 1], and [1 1 1] directions as can be seen in Fig. 1c. Due to the strong spin-orbit coupling, the ground state of the Yb 3+ (4f 13 ) ion is the total angular momentum j = 7/2 state under a spherical symmetry with a local quantization axis corresponding to one of threefold axes. The eight-fold j = 7/2 state is split into four Kramers doublets (see Supplementary Fig. 2) under the 3v symmetry, and the highest energy Kramers doublet states can be expressed by, (1) Here, | 7 2 , ⟩ is a state with j = 7/2 and = m, and is the local quantization axis at the i-th site in a tetrahedron (see Supplementary Fig. 3).
The Yb 3+ (4f 13 ) ground state corresponds to one electron removal state at this Kramers doublet from the fully occupied 4f 14 state [1]. The doublet ground state is energetically separated from the first excited state by 38.2 meV (= 443 K•kB) [2]. As the results, the Yb 3+ magnetic spin effectively has a pseudospin-1/2 with a mixture of spin 1/2 and -1/2 states. As the results, the magnetic interaction should be described by a generalized magnetic exchange Hamiltonian with tensor forms of the exchange coupling tensor ′and gtensor ′ in the local coordinates, The doublet ground state can be rewritten as | ⟩ = − ,− | ⟩ with the pseudospin 1/2 states denoted by  (= ±). Here ,− is the one electron annihilation operator to reduce to the |− ⟩ state. In this notation, the pseudospin 1/2 operators , which correspond to the Pauli spin operators on the subspace of the Kramers doublet at the i-th site, are given by Now the generalized magnetic Hamiltonian has a matrix form of ′ and ′  are site independent and respectively expressed as The local magnetic moment operator Mi is equivalent to ⊥ ( ,̂− ,̂) + ∥ ,̂ where ⊥ = 2 ⟨ + | , | − ⟩ = −2 ⟨ + | , | − ⟩ and ∥ = 2 ⟨ + | , | + ⟩ . It is noticed that the total angular momentum operators can be also expressed with the pseudospin operators like J = ⊥ ( ,̂− ,̂) + ∥ ,̂ where ⊥ = 2 ⟨ + |J , | − ⟩ = J ⊥ , ∥ = 2 ⟨ + |J , | + ⟩ = J ∥ , and the Landé g-factor gJ = 8/7. Now ℋ gen can be expressed in the global coordinates by defining transform matrices under the Td symmetry as followings, The transform matrices QZi transform local Cartesian coordinate ̂i, ′ , and ′ as ̂= T̂ , = T ′ , and = T ′ , respectively.
Then the generalized Hamiltonian is represented with the transformed exchange coupling and gfactor in the global coordinates as where ℋ = • • .

Supplementary Note 2. g-factors: Electron paramagnetic resonance
In order to determine the g-factors, we performed high-frequency electron paramagnetic resonance (EPR) measurements at ν = 104 GHz using the transmission spectrometer developed at the National High Magnetic Field Laboratory with a sweepable 15-T superconducting magnet. Supplementary Fig. 4 shows the EPR spectrum of the Ba3Yb2Zn5O11 powder sample measured at T = 295 K. The spectrum is composed of two peaks, which is well simulated with the two effective g-values, ∥ = 2.54 and ⊥ = 2.13 and the peak-to-peak linewidths, ΔHpp = 197.9(6) mT and 434.3(7) mT (see the solid red line). The observed Lorentzian line-shape means that the EPR signal is exchange-narrowed due to fast electronic fluctuations of Yb 3+ ions through an antiferromagnetic exchange interaction. We note that the experimentally determined g-values are not significantly different from those evaluated by the crystal field model calculation ∥ = 2.87 and ⊥ = 2.27 (see Supplementary Note 4).

Supplementary Note 3. Magnetization
The effective Hamiltonian ℋ eff consisting of the Heisenberg interaction, the DM interaction and the anisotropic Zeeman term (see Supplementary Note 1) enables us to explain three characteristic features of the field dependent magnetization M(H) such as step-like jumps at HC1 = 3.5 T and HC2 = 8.8 T, non-zero slope below HC1, and hysteresis near HC1. The DM interaction [5] and the anisotropic Zeeman term, which are non-commutative with the Heisenberg interaction term and then reconstruct eigenstates of the conventional Heisenberg Hamiltonian, not only affects level crossing critical fields (HC1 and HC2) but also drives a paramagnetic response to applied magnetic field below HC1. Supplementary Fig. 5 shows three simulated M(H)'s from ℋ eff under an adiabatic process, which are compared to the measured ones with the lowest rate (7.5 mT•min -1 ) in magnetic field-up and -down sweeps. No DM interaction (d/J = 0) with an isotropic g-factor (g = 2.569), no DM with anisotropic g-factors ( ∥ = 3.0 and ⊥ = 2.4), and the DM with anisotropic g-factors (d/J = 0.27, ∥ = 3.0 and ⊥ = 2.4) are represented by a black dashed, a cyan dashed and a green solid line, respectively. Among three models, the last one (the green line) gives the best simulation. It is noted that the effect of DM on the M(H) in 0 < H < HC1 is twice bigger than that of the anisotropic Zeeman term as shown in the inset of Supplementary Fig. 5, which indicates that the DM interaction admixes the eff = 0 single states and eff = 1 triplet states more significantly.
The simulated M(H) well reproduces overall features except the hysteretic behavior related to a finite field sweep rate (non-adiabatic process). This hysteretic behavior results from the non-equilibrium Landau-Zener transition, which involves a two-level system with avoided level crossing [6,7]. As the level energies vary with the magnetic field as shown in the inset of Fig. 1d, the transition probability between the two levels can be described by = 1 − exp (− Δ 0 2 4ℏ B ⁄ ) with the avoided level crossing energy gap 0 and the field sweep rate r. As the sweep rate r goes to zero (adiabatic process), the transition probability becomes 1 and the hysteretic behavior disappears since the level occupations are simply determined by the Boltzmann factor. Meanwhile, as r increases, the probability decreases and the hysteretic behavior becomes noticeable. Indeed, one can recognize that the hysteretic behavior becomes intensified in M(H) with a faster sweep rate (30 mT•min -1 ) as shown in Supplementary Fig. 6.

Supplementary Note 4. YbO6 crystal field analyses and magnetic exchange coupling constants
The crystal field (CF) splittings of Yb 4f in Ba3Yb2Zn5O11 can be estimated by using a simple point charge model under the 3v local symmetry of a distorted YbO6 octahedron. The large spin orbit coupling energy  splits the atomic 4f level into the total angular momentum J7/2 and J5/2 states. In Yb 3+ (4f 13 ), the low lying J5/2 state is fully occupied and one hole is left in the high lying J7/2 state. The splitting energy is sufficiently large in comparison with the 4f crystal field splitting energy so that we neglect the J5/2 states. The 8-fold J7/2 state is split into two doublets and a quartet by the Oh CF, and finally into four doublets (Kramers doublets) under the 3v local symmetry as schematically depicted in Supplementary Fig. 2. For estimation of the level splittings, the ionic positions are referred to the neutron powder diffraction results with Rietveld refinements. The CF Hamiltonian under the 3v symmetry can be described by six CF terms, Using the Kramers doublet states, we now examine the magnetic exchange coupling constants (i= 1, 2, 3, and 4) in the tensor as presented in Supplementary Note 1. Considering two edge-shared Yb ions at sites 3 and 4 (see Supplementary Fig. 3), the hopping Hamiltonian, which is spin-independent, is expressed as where is an annihilation operator of 4f -orbital with a  spin at the i-th site. Confining the Kramers doublet for the Yb site, we can only consider four states of the two Yb site cluster for the hopping. Those states are |Ψ 1 ⟩ = | + ⟩ 3 ⊗ | + ⟩ 4 , |Ψ 2 ⟩ = | + ⟩ 3 ⊗ | − ⟩ 4 , |Ψ 3 ⟩ = | − ⟩ 3 ⊗ | + ⟩ 4 , and |Ψ 4 ⟩ = | − ⟩ 3 ⊗ | − ⟩ 4 . In the limit of the second-order perturbation, its effective Hamiltonian is given by where |Ψ ℎ ⟩ refer to unperturbed excited states with energy Em , which are overlapped with |Ψ ⟩ through the superexchange hopping. For simplicity, we adopt a fixed U = Em -E0 for all |Ψ ℎ ⟩. Then the ℋ (34) matrix can be expressed in terms of the exchange coupling matrix elements ′ (, = x, y, z) in the local coordinates as defined in Supplementary Note 1: resulting in d/J = 0.281, which agrees well with the ratio d/J = 0.27 determined from the inelastic neutron scattering data.