Microscopic origin and quantitative spectroscopic theory of the large unquenched V$^{3+}$ orbital moment in the van der Waals ferromagnet VI$_3$

The layered van der Waals ferromagnet VI$_3$, a Mott insulator (insulating because of electron–electron repulsion rather than filled bands), exhibits an exceptionally large unquenched orbital moment on V$^{3+}$ ($m_{\mathrm{orb}}\approx 0.6\,\mu_{\mathrm{B}}$ per V from the x-ray magnetic circular dichroism orbital sum rule), a rarity among 3$d$ compounds where crystal fields normally quench orbital angular momentum. The measured spectra are best reproduced by coexistence of two inequivalent V sites with opposite trigonal distortions, while the best single-site ligand-field multiplet simulation (the high-orbital-moment one) overestimates the orbital moment by roughly a factor of two, and density-functional theory finds the high-orbital-moment and moment-quenched solutions separated by only ~5.6 meV. The open problem is to determine the microscopic V$^{3+}$ ground state of bulk VI$_3$ — in particular whether two structurally inequivalent V sites genuinely coexist and what mechanism creates them — and to construct a quantitative many-body theory of the V $L_{2,3}$-edge absorption and dichroism spectra that simultaneously reproduces the sum-rule orbital moment and the full measured line shapes, replacing the atomic multiplet description whose quantitative failure in this solid is itself unexplained.

Background

VI$_3$ is a layered van der Waals (vdW) semiconductor whose small gap is of correlation (Mott–Hubbard) rather than band origin, and it orders ferromagnetically below about 50 K and remains magnetic down to the monolayer limit, making it a prominent platform for two-dimensional magnetism. Its V ions are trivalent (3$d^2$) and sit on a honeycomb lattice inside edge-sharing octahedra of iodine ions; the compound undergoes a structural symmetry-lowering transition near 79–80 K, and additional anomalies near 36 K and 32 K have been reported. In an octahedral crystal field the five V 3$d$ orbitals split into lower $t_{2g}$ and higher $e_g$ levels; the two valence electrons occupy $t_{2g}$, which carries an effective orbital angular momentum $l=1$ rather than being fully quenched. A trigonal distortion of the octahedra further splits $t_{2g}$ into an $a_{1g}$ singlet and an $e'_g$ doublet, giving two candidate V$^{3+}$ ground states: (i) both electrons in $e'_g$ ($e'^2_g$), an orbital singlet with $L=0$ and a fully quenched orbital moment, or (ii) one electron in $a_{1g}$ and one in the spin-orbit-split $e'_g$ doublet ($a^1_{1g}e'^1_{g-}$), which sustains an unquenched orbital moment with $L=1$. Which state is realized controls the magnetocrystalline anisotropy — via Bruno-type relations that tie anisotropy energy to the orbital moment (OM) — and the size of the ordered moment, both anomalously large in VI$_3$. X-ray magnetic circular dichroism (XMCD), the difference of x-ray absorption spectra (XAS) measured with opposite circular polarizations at the V $L_{2,3}$ edges (2p→3d transitions), separates spin and orbital moments element-specifically through magneto-optical sum rules, although at the V $L_{2,3}$ edges the spin sum rule is unreliable because the $L_3$ and $L_2$ absorption edges strongly overlap. Hovančík et al. (Nano Lett. 2023) applied this to bulk VI$_3$, obtaining $m_{\mathrm{orb}}=0.6(1)\,\mu_{\mathrm{B}}$ antiparallel to the spin — direct proof of an unquenched OM — and analyzed the spectra with ligand-field multiplet simulations (atomic Hartree–Fock multiplets plus a local crystal field, including the 2p core-hole interaction). Their key findings frame the open problem. First, the simulation corresponding to the $a^1_{1g}e'^1_{g-}$ (high-OM) ground state reproduces the qualitative spectral shape but overestimates the single-site orbital moment by about a factor of two; only a 50/50 average of a high-OM site and a moment-quenched ($e'^2_g$) site, as proposed earlier from inelastic neutron scattering by Lane et al., matches both the line shapes and the sum-rule value, supporting two coexisting V sites with opposite trigonal distortions — but the microscopic origin and spatial arrangement of such sites are not established. Second, their density-functional-theory (DFT) calculations — generalized-gradient approximation with an on-site Coulomb repulsion $U$ (GGA+U), including spin-orbit coupling — converge to two nearly degenerate solutions, the high-OM one lower by only ~5.6 meV, so weak perturbations could in principle switch the balance. Third, an angle-resolved photoemission spectroscopy (ARPES) study (De Vita et al. 2022) had found substantial $a_{1g}$ occupation, attributed by its authors to V$^{2+}$ at the surface — an interpretation in tension with bulk V$^{3+}$ XAS that the high-OM ground state could instead explain. The authors explicitly identify the needed next step: the importance of many-body effects and solid-state hybridization, plus possible multiple V sites, makes VI$_3$ a challenging theory problem, and a more accurate description of the spectra may be achieved with advanced configuration-interaction techniques (explicit mixing of several electronic configurations) or the Bethe–Salpeter equation (a two-particle many-body treatment of absorption spectra including electron–hole interactions), since atomic multiplet models are local and miss band broadening, $p$-$d$ covalency, and nonlocal screening in the solid.

Problem Statement

For bulk VI$_3$, establish the microscopic V$^{3+}$ ground state and its quantitative spectroscopic description. Concretely: (1) determine whether the low-temperature phase genuinely contains two coexisting, structurally inequivalent V sites with different orbital occupations (high-orbital-moment $a^1_{1g}e'^1_{g-}$ versus moment-quenched $e'^2_g$), and if so identify the mechanism that creates them and their spatial arrangement relative to the known structural transitions — or else show that a single-site or uniform-lattice description suffices; and (2) construct a quantitative electronic-structure theory of the V $L_{2,3}$-edge XAS and XMCD — incorporating the solid-state effects absent from atomic multiplet models (interatomic hybridization and band broadening, nonlocal screening, and configuration-interaction/many-body corrections), each of which the theory must either treat explicitly or demonstrate to be negligible — that simultaneously reproduces, within the experimental uncertainty, the measured orbital sum-rule value $m_{\mathrm{orb}}=0.6(1)\,\mu_{\mathrm{B}}$ per V and the full measured line shapes of Hovančík et al. (ref. 1), thereby explaining why the best single-site multiplet simulation overestimates the orbital moment by roughly a factor of two. These two deliverables jointly constitute one objective: a validated microscopic account of how the large unquenched orbital moment of VI$_3$ arises and why it has the measured magnitude — the spectroscopic theory (2) is the quantitative test that adjudicates the ground-state assignment (1). The near-degeneracy of the high-OM and quenched-OM DFT solutions (~5.6 meV) remains essential context for this objective, since it underlies the system's observed sensitivity and the competing interpretations catalogued below.

References

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